Łukasiewicz logic and Riesz spaces

Abstract

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from $[0,1]$. Extending Mundici’s equivalence between MV-algebras and $\ell$-groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C$^*$-algebras. The propositional calculus ${\mathbb R}{\mathcal L}$ that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz $\infty$-valued propositional calculus and is complete with respect to evaluations in the standard model $[0,1]$. We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in ${\mathbb R}{\mathcal L},$ and relate them with the analogue of de Finetti’s coherence criterion for ${\mathbb R}{\mathcal L}$.

1 Introduction

MV-algebras are the algebraic structures of Łukasiewicz $\infty$-valued logic. The real unit interval $[0,1]$ equipped with the operations

$$\begin{aligned} x^*=1-x \, \mathrm{and}\, x\oplus y=\min (1,x+y) \end{aligned}$$

for any $x, y\in [0,1]$ is the standard MV-algebra, i.e. an equation holds in any MV-algebra if and only if it holds in $[0,1]$. Mundici (1995) proved that MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit. Consequently, for any MV-algebra there exists a lattice-ordered group with strong unit $(G,u)$ such that $A\simeq [0,u]_G$, where

$$\begin{aligned}&{[0,u]_G}=([0,u],\oplus ,^{*},0),\\&{[0,u]}=\{x\in G \mid 0\le x\le u\}, \\&x\oplus y=(x+y)\wedge u \,\,\mathrm{and}\,\, x^{*}=u-x \,\,\mathrm{for\, any}\,\, x, y\in [0,u]. \end{aligned}$$

If $(V,u)$ is a Riesz space (vector-lattice) (Luxemburg and Zaanen 1971) with strong unit then the unit interval $[0,u]_V$ is closed to the scalar multiplication with scalars from $[0,1]$. The structure

$$\begin{aligned}{}[0,u]_V=([0,u],\cdot ,\oplus ,^*,0), \end{aligned}$$

where $([0,u],\oplus ,^*,0)$ is the MV-algebra defined as above and $\cdot :[0,1]\times [0,u]_V\rightarrow [0,u]_V$ that satisfies the axioms of the scalar product is the fundamental example in the theory of Riesz MV-algebras, initiated in Di Nola and Leuştean (2011) and further developed in the present paper.

The study of Riesz MV-algebras is related to the problem of finding a complete axiomatization for the variety generated by $([0,1],\cdot ,\oplus ,^*,0)$, where $([0,1],\oplus ,^*,0)$ is the standard MV-algebra and $\cdot$ is the product of real numbers. The investigations led to the definition of product MV-algebras (PMV-algebras), which can be represented as unit intervals in lattice-ordered rings with strong unit (Di Nola and Dvurečenskij 2001). A PMV-algebra is a structure $(P,\cdot )$, where $P$ is an MV-algebra and $\cdot :P\times P\rightarrow P$ satisfies the equations of an internal product. PMV-algebras are an equational class but, as a variety, they are not generated by [0, 1]. The quasi-variety of PMV-algebras generated by [0, 1] is axiomatized in Montagna (2005). In this context, it was natural to replace the internal product with an external one: a Riesz MV-algebra is a structure $(R,\cdot )$, where $R$ is an MV-algebra and $\cdot :[0,1]\times R\rightarrow R$. Since we prove that the variety of Riesz MV-algebras is generated by $[0,1]$, the propositional calculus ${\mathbb R}\mathcal{L}$, that has Riesz MV-algebras as models, is complete with respect to evaluations in $[0,1]$.

The Riesz MV-algebras are defined in Di Nola and Leuştean (2011). In Sect. 3 we give an equivalent but more suitable definition of these structures and we prove some of their fundamental properties.

The categorical equivalence between Riesz MV-algebras and Riesz spaces with strong unit is proved in Sect. 4. As a consequence, the standard Riesz MV-algebra $[0,1]$ generates the variety of Riesz MV-algebras.

In Sect. 5, the categorical equivalence is specialized to the class of norm-complete Riesz MV-algebras, which is dually equivalent to the category of compact Hausdorff spaces. Using the Gelfand–Naimark duality, we establish a connection with the theory of commutative unital C$^*$-algebras.

Section 6 presents the propositional calculus ${\mathbb R}\mathcal{L}$ which simplifies the one introduced in Di Nola and Leuştean (2011). In Sect. 7 we prove a normal form theorem for formulas of ${\mathbb R}\mathcal{L}$. Since ${\mathbb R}\mathcal{L}$ is a conservative extension of Łukasiewicz logic $\mathcal L$, this theorem is a generalization of McNaughton theorem (McNaughton 1951). Our result asserts that for any continuous piecewise linear function $f:[0,1]^n\rightarrow [0,1]$ there exists a formula $\varphi$ of ${\mathbb R}\mathcal{L}$ with $n$ variables such that $f$ is the term function associated to $\varphi$.

In Sect. 8 we initiate the theory of quasi-linear combinations of formulas in ${\mathbb R}\mathcal{L}$. If $f_i:[0,1]^n\rightarrow {\mathbb R}$ are continuous piecewise linear functions and $c_i$ are real numbers for any $i\in \{1,\ldots ,k\}$, then the normal form theorem guarantees the existence of a formula $\Phi$ of ${\mathbb R}\mathcal{L}$, whose term function is equal to $((\sum _{i=1}^kc_if_i)\vee 0)\wedge 1,$ and in this case, we say that $\Phi$ is quasi-linear combination of $f_1,\ldots , f_k$. We prove de Finetti’s coherence criterion for ${\mathbb R}\mathcal{L}$ and provide an equivalent characterization by the fact that a quasi-linear span contains only invalid formulas.

Some of results contained in this paper may overlap with the proceeding paper Di Nola and Leuştean (2011). Other results are proved in a more general setting in Di Nola et al. (2003); Flondor and Leuştean (2004); Leuştean (2004). For the sake of completeness we sketched the proofs that we consider important for the present development.

2 Preliminaries on MV-algebras

An MV-algebra is a structure $(A,\oplus ,^{*},0)$, where $(A,\oplus , 0)$ is an abelian monoid and the following identities hold for all $x,y\in A$:

• (MV1) $(x^{*})^{*}=x$,

• (MV2) $0^{*}\oplus x=0^{*}$,

• (MV3) $(x^{*}\oplus y)^{*}\oplus y=(y^{*}\oplus x)^{*}\oplus x$.

We refer to Cignoli et al. (2000) for all the unexplained notions concerning MV-algebras and to Mundici (2011) for advanced topics. On any MV-algebra $A$ the following operations are defined for any $x,y\in A$:

$$\begin{aligned}&1=0^*, x\odot y=(x^*\oplus y^*)^*, x\rightarrow y=x^*\oplus y\\&0x=0, mx=(m-1)x\oplus x \,\,\mathrm{for\, any}\,\, m\ge 1. \end{aligned}$$

We assume in the sequel that the operation $\odot$ is more binding then $\oplus$.

Remark 1

Any MV-algebra $A$ is a bounded distributive lattice, with the partial order defined by

$$\begin{aligned} x\le y \,\,\mathrm{if\, and\, only\, if}\, x\odot y^*=0 \end{aligned}$$

and the lattice operations defined by

$$\begin{aligned} x\vee y=x\oplus y\odot x^* \,\mathrm{and}\, x\wedge y=x\odot (x^*\oplus y) \end{aligned}$$

for any $x$, $y\in A$.

Any MV-algebra $A$ has an internal distance:

$$\begin{aligned} d(x,y)=(x\odot y^*)\oplus (x^*\odot y) \,\,\mathrm{for\, any}\,\, x, y\in A. \end{aligned}$$

Lemma 1

((Cignoli et al. 2000, Proposition 1.2.5)) In any MV-algebra $A$, the following properties hold for any $x$, $y$, $z\in A$:

• (a) $d(x,y)=d(y,x)$,

• (b) $d(x,y)=0$    iff $x=y$,

• (c) $d(x,z)\le d(x,y)\oplus d(y,z).$

If $(A,\oplus ,^*,0)$ is an MV-algebra then an ideal is a nonempty subset $I\subseteq A$ such that for any $x$, $y\in A$ the following conditions are satisfied:

• (i1) $x\in I$ and $y\le x$ imply $y\in I$,

• (i2) $x$ and $y\in I$ imply $x\oplus y\in I$.

An ideal $I$ of $A$ uniquely defines a congruence $\sim _I$ by

$$\begin{aligned} x\sim _I y \,\,\mathrm{iff}\,\, x\odot y^*\in I \,\,\mathrm{and}\,\, y\odot x^*\in I. \end{aligned}$$

We denote by $A/I$ the quotient MV-algebra and we refer to Cignoli et al. (2000) for more details.

We recall that an $\ell$ -group is a structure $(G,+,0,\le )$ such that $(G,+,0)$ is a group, $(G,\le )$ is a lattice and any group translation is isotone (Bigard et al. 1977). In the following the $\ell$-groups are abelian. For an $\ell$-group $G$ we denote $G_+=\{x\in G\mid x\ge 0\}$. An element $u\in G$ is a strong unit if $u\ge 0$ and for any $x\in G$ there is a natural number $n$ such that $x\le nu$. An $\ell$-group is unital if it posses a strong unit. If $(G,u)$ is a unital $\ell$-group, we define $[0,u]=\{x\in G\mid 0\le x\le u\}$ and

$$\begin{aligned} x\oplus y=(x+y)\wedge u, x^*=u-x \quad \mathrm{for\, any}\,\, x, y\in [0,u]. \end{aligned}$$

Then $[0,u]_G=([0,u],\oplus ,\lnot ,0)$ is an MV-algebra ((Cignoli et al. 2000, Proposition 2.1.2)).

Lemma 2

((Cignoli et al. 2000, Lemma 7.1.3)) Let $(G,u)$ be a unital $\ell$-group, $x\ge 0$ in $G$ and $n\ge 1$ a natural number such that $x\le nu$. Then $x=x_1+\cdots + x_n$, where

$$\begin{aligned} x_i=((x-(i-1)u)\vee 0)\wedge u\in [0,u] \end{aligned}$$

for any $i\in \{1,\ldots ,n\}$.

We denote by ${\mathbb M}{\mathbb V}$ the category of MV-algebras and by ${\mathbb A}{\mathbb G}_u$ the category of unital abelian lattice-ordered groups with unit-preserving morphisms. In Mundici (1986) the functor ${{\mathrm{\Gamma }}}:{\mathbb A}{\mathbb G}_u\rightarrow {\mathbb M}{\mathbb V}$ is defined as follows:

${{\mathrm{\Gamma }}}(G,u)=[0,u]_G$ for any unital $\ell$-group $(G,u)$,

${{\mathrm{\Gamma }}}(f)=f|_{[0,u]}$ for any morphsim $f:(G,u)\rightarrow (G^{\prime },u^\prime )$ from ${\mathbb A}{\mathbb G}_u$.

Theorem 1

((Cignoli et al. 2000, Corollary 7.1.8)) The functor ${{\mathrm{\Gamma }}}$ yields an equivalence between ${\mathbb A}{\mathbb G}_u$ and ${\mathbb M}{\mathbb V}$.

Definition 1

If $A$ and $B$ are MV-algebras then a function $\omega :A\rightarrow B$ is called additive if

$$\begin{aligned}&x\odot y=0\,\, \mathrm{implies}\,\, \omega (x)\odot \omega (y)=0 \,\,\mathrm{and}\,\, \omega (x\oplus y)\\&\quad =\omega (x)\oplus \omega (y). \end{aligned}$$

Additivity was firstly studied in the context of states defined on MV-algebras Mundici (1995). The theory of states generalizes the boolean probability theory and reflects the theory of states defined on $\ell$-groups.

Definition 2

(Mundici 1995) If $A$ is an MV-algebra then a function $s:A\rightarrow [0,1]$ is a state if the following properties are satisfied for any $x$, $y\in A$:

• (s1) if $x\odot y=0$ then $s(x\oplus y)=s(x)+s(y)$,

• (s2) $s(1)=1$.

The following results are proved in Leuştean (2004), but we sketch the proofs for the sake of completeness. We also note that particular instances of these results are proved in Flondor and Leuştean (2004). Proposition 1 is proved for states in Mundici (1995).

Proposition 1

Assume $(G,u)$ and $(H,v)$ are unital $\ell$-groups, $A={{\mathrm{\Gamma }}}(G,u)$ and $B={{\mathrm{\Gamma }}}(H,v)$. Then for any additive function $\omega :A\rightarrow B$ there exists a unique group morphism $\overline{\omega }:G\rightarrow H$ such that $\overline{\omega }(x)=\omega (x)$ for any $x\in [0,u]$.

Proof

If $x\in G$ and $x\ge 0$ then there are $x_{1}$, $\ldots$, $x_{m}\in [0,u]$ such that $x=x_{1}+\cdots +x_{m}$. Then we define

$$\begin{aligned} \overline{\omega }(x):=\omega (x_{1})+\cdots +\omega (x_{m}). \end{aligned}$$

The fact that $\overline{\omega }(x)$ is well defined follows by Riesz decomposition property in $\ell$-groups ((Bigard et al. 1977, 1.2.16)). Hence $\overline{\omega }(x)$ is well defined for $x\in G_{+}$ and $\overline{\omega }(x+y)=\overline{\omega }(x)+\overline{\omega }(y)$ for any $x$, $y\in G_{+}$. By ((Bigard et al. 1977, 1.1.7)) it follows that $\overline{\omega }$ can be uniquely extended to a group homomorphism defined on $G$. $\square$

Lemma 3

If $A$ and $B$ are MV-algebras and $\omega :A\rightarrow B$ is a function, then the following are equivalent:

• (a) $\omega$ is additive,

• (b) the following properties hold for any $x$, $y\in A$:

  • (b1) $x\le y$ implies $\omega (x)\le \omega (y)$,

  • (b2) $\omega (x\odot (x\wedge y)^{*})=\omega (x)\odot \omega (x\wedge y)^{*}$.

Proof

(a)$\Rightarrow$(b) If $x\le y$ then $y=x\vee y=x\oplus y\odot x^*$, so $\omega (y)=\omega (x)\oplus \omega (y\odot x^*)$ and $\omega (x)\le \omega (y)$. Hence, $\omega$ is isotone. We remark that $\omega (x\wedge y)\le \omega (x)$, so

$$\begin{aligned}&\omega (x)\odot \omega (x\wedge y)^{*}\oplus \omega (x\wedge y) = \omega (x)\vee \omega (x\wedge y)\\&\quad =\omega (x) =\omega (x\vee (x\wedge y))\\&\quad =\omega (x\odot (x\wedge y)^{*})\oplus \omega (x\wedge y). \end{aligned}$$

It follows that

$$\begin{aligned}&\omega (x)\odot \omega (x\wedge y)^{*}\\&\quad =(\omega (x)\odot \omega (x\wedge y)^{*})\wedge \omega (x\wedge y)^*\\&\quad =(\omega (x)\odot \omega (x\wedge y)^{*}\oplus \omega (x\wedge y))\odot \omega (x\wedge y)^*\\&\quad =(\omega (x\odot (x\wedge y)^{*})\oplus \omega (x\wedge y))\odot \omega (x\wedge y)^*\\&\quad =\omega (x\odot (x\wedge y)^{*})\wedge \omega (x\wedge y)^*\\&\quad =\omega (x\odot (x\wedge y)^{*}) \end{aligned}$$

(b)$\Rightarrow$(a) We remark that for $x=1$ in (b2) we get $\omega (y^{*})=\omega (1)\odot \omega (y)^{*}$, so $\omega (y^{*})\le \omega (y)^{*}$. Assume $x\odot y=0$, so $x\le y^{*}$. Using (b1), we get $\omega (x)\le \omega (y^{*})\le \omega (y)^{*}$, so $\omega (x)\odot \omega (y)=0$. In this case, using (b2) we get

$$\begin{aligned} \omega (x)\!=\!\omega (x\wedge y^{*})\!=\!\omega ((x\oplus y)\odot y^{*})\!=\!\omega (x\oplus y)\odot \omega (y)^{*}. \end{aligned}$$

It follows that:

$$\begin{aligned} \omega (x)\oplus \omega (y)&= \omega (x\oplus y)\odot \omega (y)^{*}\oplus \omega (y)\\&= \omega (x\oplus y)\vee \omega (y). \end{aligned}$$

Using (b1), $\omega (y)\le \omega (x\oplus y),$ and we get $\omega (x\oplus y)=\omega (x)\oplus \omega (y)$. $\square$

3 Riesz MV-algebras

Riesz MV-algebras are introduced in Di Nola and Leuştean (2011). Below we give a simpler and more suitable definition, which provides directly an equational characterization. The equivalence between this definition and the one from Di Nola and Leuştean (2011) is proved in Theorem 2.

Definition 3

A Riesz MV-algebra is a structure

$$\begin{aligned} (R, \cdot ,\oplus ,^*,0), \end{aligned}$$

where $(R,\oplus ,^*,0)$ is an MV-algebra and the operation $\cdot :[0,1]\times R\rightarrow R$ satisfies the following identities for any $r$, $q\in [0,1]$ and $x$, $y\in R$:

• (RMV1) $r\cdot (x\odot y^{*})=(r\cdot x)\odot (r\cdot y)^{*}$,

• (RMV2) $(r\odot q^{*})\cdot x=(r\cdot x)\odot (q\cdot x)^{*}$,

• (RMV3) $r\cdot (q\cdot x)=(rq)\cdot x$,

• (RMV4) $1\cdot x=x$.

In the following we write $rx$ instead of $r\cdot x$ for $r\in [0,1]$ and $x\in R$. Note that $rq$ is the real product for any $r$, $q\in [0,1]$.

Example 1

If $X$ is a compact Hausdorff space then

$$\begin{aligned} C(X)_u=\{f:X\rightarrow [0,1]\mid f\,\, \text{ continuous }\} \end{aligned}$$

is a Riesz MV-algebra, with all the operations defined componentwise. This example will be further investigated in Sect. 5

Example 2

If $G$ is an abelian $\ell$-group, then $R={{\mathrm{\Gamma }}}({\mathbb R}\times _{lex} G, (1,0))$ is a Riesz MV-algebra, where ${\mathbb R}\times _{lex} G$ is the lexicographic product of $\ell$-groups and the scalar multiplication is defined by $r(q,x)=(rq, x)$ for any $r\in [0,1]$ and $(q,x)\in R$.

Lemma 4

In any Riesz MV-algebra $R$ the following properties hold for any $r$, $q\in [0,1]$ and $x$, $y\in R$:

• (a) $0x=0$, $r0=0$,

• (b) $x\le y$ implies $rx\le ry$,

• (c) $r\le q$ implies $rx\le qx$,

• (d) $rx\le x$.

Proof

(a) follows by (RMV1) and (RMV2) for $x=y$ and, respectively, $r=q$.

(b), (c) follow by Remark 1.

(d) follows by (c) and (RMV4). $\square$

Proposition 2

The function $\iota :[0,1]\rightarrow R$ defined by $\iota (r)=r1$ for any $r\in [0,1]$ is an embedding. Consequently, any Riesz MV-algebra $R$ contains a subalgebra isomorphic with $[0,1]$.

Proof

By Lemma 4 we get $\iota (0)=0$. If $r$, $q\in [0,1]$ then

$$\begin{aligned}&\iota (r^*)=r^* 1=(1\cdot 1)\odot (r1)^*=(r1)^*,\\&\quad \iota (r\odot q)=\iota (r\odot q^{**})= (r\odot q^{**})1=(r1)\odot (q^* 1)^*\\&\qquad =(r1)\odot (q1)^{**}=(r1)\odot (q1). \end{aligned}$$

$\square$

A Riesz space (vector lattice) (Luxemburg and Zaanen 1971) is a structure

$$\begin{aligned} (V,\cdot , +,0,\le ) \end{aligned}$$

such that $(V,+,0,\le )$ is an abelian $\ell$-group, $(V,\cdot ,+,0)$ is a real vector space, and in addition,

$$\begin{aligned}&\mathrm{(RS)}\, x\le y\,\, \mathrm{implies}\,\, r \cdot x\le r\cdot y ,\\&\quad \mathrm{for\, any}\,\, x, y\in V\,\, \mathrm{and}\,\, r\in {\mathbb R}, r\ge 0.\\ \end{aligned}$$

A Riesz space is unital if the underlaying $\ell$-group is unital.

Lemma 5

If $(V,u)$ is a unital Riesz space, then

$$\begin{aligned}{}[0,u]_V=([0,u],\cdot ,\oplus , ^*,0) \end{aligned}$$

is a Riesz MV-algebra, where $rx$ is the scalar multiplication of $V$ for any $r\in [0,1]$ and $x\in [0,u]$.

Proof

Assume $r, q\in [0,1]$ and $x, y\in [0,u]$.

• (RMV1) $r(x\odot y^*)=r((x-y)\vee 0)=(r x -r y)\vee 0=(r x)\odot (r y)^*$.

• (RMV2) If $r\le q$ then $r x \le q x$, so $(r\odot q^*)x=$ $((r-q)\vee 0)x=0=(rx-qx)\vee 0=(r x)\odot ( q x)^*$.

If $r> q$ then $((r-q)\vee 0)x=(r-q)x=rx-q x =(r x)-(q x)\vee 0=(r x)\odot ( q x)^*$.

We note that (RMV3) and (RMV4) hold in $V$, therefore they hold in $[0,u]$. $\square$

Remark 2

If $(R, \cdot ,\oplus ,^*,0)$ is a Riesz MV-algebra then we denote its MV-algebra reduct by ${{\mathrm{U}}}(R)=(R,\oplus ,^*,0)$. Assume $I$ is an ideal of ${{\mathrm{U}}}(R)$. By Lemma 4 (d) we infer that $rx\in I$ whenever $r\in [0,1]$ and $x\in I$. It follows, by (RMV1), that $rx\sim _I ry$ whenever $r\in [0,1]$ and $x\sim _I y$. As consequence, the quotient $R/I$ has a canonical structure of Riesz MV-algebra.

Remark 3

A Riesz MV-algebra $R$ has the same theory of ideals (congruences) as its reduct ${{\mathrm{U}}}(R)$. If $R$ is a Riesz MV-algebra and $P\subseteq R$ an ideal then it is straightforward that the following hold:

• (a) $P$ is prime iff $R/P$ is linearly ordered,

• (b) $P$ is maximal iff $R/P\simeq [0,1]$.

Note that (b) holds since, for any maximal ideal $P$, the quotient $R/P$ is an MV-subalgebra of $[0,1]$. But the only subalgebra of $[0,1]$ which is a Riesz MV-algebra is $[0,1]$ by Proposition 2, so $R/P\simeq [0,1]$.

Lemma 6

If $R$ is a Riesz MV-algebra, $I\subseteq R$ an ideal and $x\in R$ such that $rx\in I$ for some $r\in (0,1]$ then $x\in I$.

Proof

Let $r\in (0,1]$ such that $rx\in I$ and let $m$ be the integer part of $\frac{1}{r}$. Hence $\frac{1}{m+1}x\le rx$, so $\frac{1}{m+1}x\in I$. Since $x=(m+1)\left( \frac{1}{m+1}x\right)$ we get $x\in I$. $\square$

Corollary 1

Any simple Riesz MV-algebra is isomorphic with $[0,1]$. Any semisimple Riesz MV-algebra is a subdirect product of copies of $[0,1]$.

In the sequel we investigate the morphisms of Riesz MV-algebras.

Corollary 2

If $R_1$ and $R_2$ are Riesz MV-algebras and $f:U(R_1)\rightarrow U(R_2)$ is a morphism of MV-algebras then

$$\begin{aligned} f(rx)=rf(x)\,\, \mathrm{for\, any}\,\, r\in [0,1] \,\,\mathrm{and}\,\, x\in R_1. \end{aligned}$$

Proof

Assume $J$ is an ideal in $R_2$. Since $f$ is an morphism of MV-algebras, it follows that $f^{-1}(J)$ is an ideal in $R_1$. If $x\in R_1$ and $r\in [0,1]$ we have

$$\begin{aligned}&r f(x)\in J\Rightarrow f(x)\in J\Rightarrow x\in f^{-1}(J) \Rightarrow \\&\quad r x \in f^{-1}(J)\Rightarrow f(r x)\in J,\\&f(r x)\in J\Rightarrow r x \in f^{-1}(J) \Rightarrow x \in f^{-1}(J) \Rightarrow \\&\quad f(x) \in J \Rightarrow r f(x) \in J. \end{aligned}$$

Note that we used Lemma 6 twice. We proved that, for any ideal $J$ of $R_2$

$$\begin{aligned} r f(x)\in J \Leftrightarrow f(r x)\in J. \end{aligned}$$

Therefore, $rf(x)\odot f(rx)^*\in J$ and $f(rx)\odot (rf(x))^*$ for any ideal $J$ of $R_2$. This means that $rf(x)\odot f(rx)^*=f(rx)\odot (rf(x))^*=0,$ so $f(rx)=rf(x).$ $\square$

Remark 4

The above result asserts that a morphism of Riesz MV-algebra is simply a morphism between the corresponding MV-algebra reducts.

The following result is similar with Chang’s representation theorem for MV-algebras ((Chang 1959, Lemma 3)).

Corollary 3

Any Riesz MV-algebra is a subdirect product of linearly ordered Riesz MV-algebras.

Proof

If $R$ is a Riesz MV-algebra then, by Remark 3, $\bigcap \{P\mid P \text{ prime } \text{ ideal } \text{ of } R\}=\{0\}$ and $R/P$ is linearly ordered for any prime ideal $P$. As consequence, $R$ is a subdirect product of the family

$$\begin{aligned} \{R/P\mid P \text{ prime } \text{ ideal } \text{ of } R\}. \end{aligned}$$

$\square$

To prove that Riesz MV-algebras introduced in Definition 3 coincide with the ones defined in Di Nola and Leuştean (2011), we recall the following.

Remark 5

(Flondor and Leuştean 2004) If $\Omega$ is a set of unary operation symbols, then an MV-algebra with $\Omega$ -operators is a structure $(A,\Omega _A)$ where $A$ is an MV-algebra and for any $\omega \in \Omega$ the operation $\omega _A:A\rightarrow A$ is additive. An additive function $\omega : A\rightarrow A$ is an $f$ -operator if

$$\begin{aligned} x\wedge y=0\,\, \mathrm{implies}\,\, \omega (x)\wedge y=0\,\, \mathrm{for any}\,\, x, y\in A. \end{aligned}$$

If $(A,\Omega _A)$ is an MV-algebra with $\Omega$-operators such that $\omega _A$ is an $f$-operator for any $\omega \in \Omega$, then $(A,\Omega _A)$ is a subdirect product of linearly ordered MV-algebras with $\Omega$-operators ((Flondor and Leuştean 2004, Corollary 5.6.)).

Remark 6

Assume that $(R,\oplus ,^*,0)$ is an MV-algebra and let $\cdot :[0,1]\times R\rightarrow R$ such that (RMV2), (RMV3), (RMV4) hold and the function

$$\begin{aligned} \omega _r:R\rightarrow R, \omega _r(x)=r\cdot x \end{aligned}$$

is additive for any $r\in [0,1]$. By (RMV2) and (RMV4) we get $\omega _r(x)\le x$ for any $r\in [0,1]$ and $x\in R$, so $\omega _r$ is an $f$-operator for any $r\in [0,1]$. If $\Omega =\{\omega _r\mid r\in [0,1]\}$ then, by Remark 5, $(R,\Omega )$ is an MV-algebra with $\Omega$-operators that can be represented as subdirect product of linearly ordered MV-algebras with $\Omega$-operators.

Lemma 7

Assume that $(R,\oplus ,^*,0)$ is an MV-algebra. If $\cdot :[0,1]\times R\rightarrow R$ then the following are equivalent:

(RMV2) $(r\odot q^{*})\cdot x=(r\cdot x)\odot (q\cdot x)^{*}$

for any $r$, $q\in [0,1]$ and $x\in R$,

(RMV2$^\prime$) $r\odot q=0$ then $(r\cdot x)\odot (q\cdot x)=0$ and

$$\begin{aligned} (r\oplus q)\cdot x=(r\cdot x)\oplus (q\cdot x) \end{aligned}$$

for any $r$, $q\in [0,1]$ and $x\in R$.

Proof

For $x\in R$ define $\omega _x: [0,1]\rightarrow R$ by $\omega _x(r)=rx$ for any $r\in [0,1]$. If $\omega _x$ satisfies (RMV2) then the condition (b) from Lemma 3 is satisfied, so $\omega _x$ satisfies also (RMV2$^\prime$). Conversely, if $\omega _x$ satisfies (RMV2$^\prime$) then, by Lemma 3, we also get $\omega _x(0)=0$. Assume $r$, $q\in [0,1]$ such that $r\le q$. Hence $r\odot q^*=0$ and $rx\le qx$, so $(r\odot q^*)x=0x=0=(rx)\odot (qx)^*$. If $r$, $q\in [0,1]$ such that $r>q$ then (RMV2) coincides with the equation (b2) from Lemma 3. $\square$

Theorem 2

Assume that $(R,\oplus ,^*,0)$ is an MV-algebra and $\cdot :[0,1]\times R\rightarrow R$. Then $(R,\cdot , \oplus ,^*,0)$ is a Riesz MV-algebra if and only if the following properties are satisfied for any $x$, $y\in R$ and $r$, $q\in [0,1]$:

• (RMV1$^\prime$) if $x\odot y=0$ then $(r\cdot x)\odot (r\cdot y)=0$ and

  $$\begin{aligned} r\cdot (x\oplus y)=(r\cdot x)\oplus (r\cdot y), \end{aligned}$$

• (RMV2$^\prime$) if $r\odot q=0$ then $(r\cdot x)\odot (q\cdot x)=0$ and

  $$\begin{aligned} (r\oplus q)\cdot x=(r\cdot x)\oplus (q\cdot x), \end{aligned}$$

• (RMV3) $r\cdot (q\cdot x)=(rq)\cdot x$,

• (RMV4) $1\cdot x=x$.

Proof

By Lemma 7, if $(R,\oplus ,^*,0)$ is an MV-algebra and $\cdot :[0,1]\times R\rightarrow R$, then the algebra $(R,\cdot , \oplus ,^*,0)$ satisfies (RMV2), (RMV3) and (RMV4) if and only if it satisfies (RMV2$^\prime$), (RMV3) and (RMV4). Assume now that $(R,\cdot , \oplus ,^*,0)$ satisfies (RMV2), (RMV3) and (RMV4). We have to prove that (RMV1) is satisfied if and only if (RMV1$^\prime$) is satisfied. By Corollary 3 and Remark 6, it suffices to prove the equivalence for linearly ordered structures. In this case, by Lemma 3, the equivalence of (RMV1) and (RMV1$^\prime$) is straightforward. $\square$

Note that in Di Nola and Leuştean (2011) a Riesz MV-algebra is defined by (RMV1$^\prime$), (RMV2$^\prime$), (RMV3) and (RMV4), so we proved that Definition 3 is equivalent to the initial one.

4 Riesz MV-modules and Riesz spaces. The completeness theorem

By Theorem 2, Riesz MV-algebras are exactly the MV-modules Di Nola et al. (2003) over $[0,1]$. Hence some basic properties follow from the general theory of MV-modules developed in Di Nola et al. (2003); Leuştean (2004). One of the most important results is the categorical equivalence between Riesz MV-algebras and unital Riesz spaces. For the sake of completeness, we sketch the proof of this result.

Proposition 3

For any Riesz MV-algebra $R$ there is a unital Riesz space $(V,u)$ such that $R\simeq [0,u]_V$.

Proof

By Theorem 1, there exists a unital $\ell$-group $(V,u)$ such that $R$ and $[0,u]_V$ are isomorphic MV-algebras. For any $\lambda \in {\mathbb R}$ and $x\in V$ we have to define the scalar multiplication $\lambda x$. We can safely assume that $R=[0,u]\subseteq V$.

If $r\in [0,1]$ then $x\mapsto r x$ is an additive function from $[0,u]_V$ to $[0,u]_V$ so, by Proposition 1, it can be uniquely extended to a group morphism $\omega _r:V\rightarrow V$. Hence we define $r x=\omega _r(x)$ for any $x\in V$. We note that $x\ge 0$ implies $r x\ge 0$.

If $q\in [0,1]$ then $\omega _{rq}=\omega _r\circ \omega _q$ since they coincide on the positive cone, so $r(q v)=(rq)v$.

Note that $v=(v\vee 0)-((-v)\vee 0)$, so

$$\begin{aligned} 1v=1(v\vee 0)-1((-v)\vee 0)=(v\vee 0)-((-v)\vee 0)=v. \end{aligned}$$

If $\lambda \ge 0$ and $v\in V$, then there are $r_{1}$, $\ldots$, $r_{m}\in [0,1]$ such that $\lambda =r_{1}+\cdots +r_{m}$. Then we define

$$\begin{aligned} \lambda v = r_1 v+\cdots +r_m v. \end{aligned}$$

One can prove that $\lambda v$ is well defined using the Riesz decomposition property ((Bigard et al. 1977, 1.2.16)). If $\mu \ge 0$, then $\mu =q_{1}+\cdots +q_{n}$ for some $q_{1}$, $\ldots$, $q_{n}\in [0,1]$ and

$$\begin{aligned} \lambda (\mu v)&= \lambda \left( \sum _{j=1}^{n}q_j v\!\right) \!=\!\sum _{i=1}^{m}r_i\!\left( \sum _{j=1}^{n}q_j v\!\right) \!=\!\sum _{i=1}^{m}\sum _{j=1}^{n}r_i(q_j v)\\&= \sum _{i=1}^{m}\sum _{j=1}^{n}(r_iq_ j)v\!=\! \left( \sum _{i=1}^{m}\sum _{j=1}^{n}(r_iq_ j)\right) v\!=\!(\lambda \mu ) v. \end{aligned}$$

If $\lambda \le 0$ in ${\mathbb R}$ then we set $\lambda v=-(|\lambda |v))$, where $|\lambda |$ is the module of $\lambda$ in ${\mathbb R}$. It is straightforward that $\lambda (\mu v)=(\lambda \mu ) v$ for another $\mu \in {\mathbb R}$.

We know that $(V,u)$ is a unital $\ell$-group and we defined the scalar product $\lambda v$ for any $\lambda \in R$ and $v\in V$ such that $\lambda v\ge 0$ whenever $\lambda \ge 0$ and $v\ge 0$. Therefore, $(V,u)$ is a unital vector lattice. $\square$

We denote by ${\mathbb R}{\mathbb M}{\mathbb V}$ the category of Riesz MV-algebras and by ${\mathbb R}{\mathbb S}_u$ the category of unital Riesz spaces with unit-preserving morphisms.

Following this construction we get a functor

$$\begin{aligned} {{\mathrm{\Gamma _{\mathbb R}}}}:{\mathbb R}{\mathbb S}_u\rightarrow {\mathbb R}{\mathbb M}{\mathbb V} \end{aligned}$$

defined as follows:

${{\mathrm{\Gamma _{\mathbb R}}}}(V,u)=[0,u]_V$ for any unital Riesz space $(V,u)$,

${{\mathrm{\Gamma _{\mathbb R}}}}(f)=f|_{[0,u]}$ for any morphism $f:(V,u)\rightarrow (V^{\prime },u^\prime )$ from ${\mathbb R}{\mathbb S}_u$.

Theorem 3

(Di Nola and Leuştean 2011; Di Nola et al. 2003) The functor ${{\mathrm{\Gamma _{\mathbb R}}}}$ yields an equivalence between ${\mathbb R}{\mathbb S}_u$ and ${\mathbb R}{\mathbb M}{\mathbb V}$.

Proof

It follows from Theorem 1, Corollary 2 and Proposition 3. $\square$

It is straightforward that the following diagram is commutative, where ${{\mathrm{U}}}$ are forgetful functors:

The standard Riesz MV-algebra is $([0,1],\cdot ,\oplus ,^*,0)$, where $\cdot :[0,1]\times [0,1]\rightarrow [0,1]$ is the product of real numbers and $([0,1],\oplus ,^*,0)$ is the standard MV-algebra. In the sequel we prove that the variety of Riesz MV-algebras is generated by $[0,1]$, i.e. an identity holds in any Riesz MV-algebra if and only if it holds in the standard Riesz MV-algebra $[0,1]$. Our approach follows closely the proof of Chang’s completeness theorem for Łukasiewicz logic (Chang 1959). To any sentence in the first-order theory of Riesz MV-algebras we associate a sentence in the first-order theory of Riesz spaces such that the satisfiability is preserved by the ${{\mathrm{\Gamma _{\mathbb R}}}}$ functor. The first-order theory of Riesz MV-algebras, as well as the theory of Riesz spaces, is obtained considering for each scalar $r$ an unary function $\rho _r$ which denotes in a particular model the scalar multiplication by $r$, i.e. $x\mathop {\mapsto }\limits ^{\rho _r} rx$. In the following, the language of Riesz MV-algebras is ${\mathcal L}_\mathrm{RMV}=\{\oplus ,^*,0,\{\rho _r\}_{r\in [0,1]}\}$ and the language of Riesz spaces is ${\mathcal L}_\mathrm{Riesz}=\{\le , +,-,\vee ,\wedge , 0,\{\rho _r\}_{r\in \mathbb {R}}\}$.

Let $t(v_{1},\ldots ,v_{k})$ be a term of ${\mathcal L}_{RMV}$ and $v$ a propositional variable different from $v_{1}$, $\ldots$, $v_{k}$. We define $\overline{t}$ as follows:

• if $t=0$ then $\overline{0}$ is $0$,

• if $t=v$ then $\overline{t}$ is $v$

• if $t=t_{1}^*$ then $\overline{t}$ is $v-\overline{t_{1}}$,

• if $t=t_{1}\oplus t_{2}$ then $\overline{t}$ is $(t_{1}+t_{2})\wedge v$,

• if $t=\rho _r(t_1)$ then $\overline{t}$ is $\rho _r(\overline{t_1})$.

Let $\varphi (v_{1},\ldots ,v_{k})$ be a formula of ${\mathcal L}_{RMV}$ such that all the free and bound variables of $\varphi$ are in $\{v_{1},\ldots ,v_{k}\}$ and $v$ a propositional variable different from $v_{1}$, $\ldots$, $v_{k}$. We define $\overline{\varphi }$ as follows:

• if $\varphi$ is $t_{1}=t_{2}$ then $\overline{\varphi }$ is $\overline{t_1}=\overline{t_2}$,

• if $\varphi$ is ${\lnot }\psi$ then $\overline{\varphi }$ is ${\lnot }\overline{\psi }$,

• if $\varphi$ is $\psi {\vee }\chi$ then $\overline{\varphi }$ is $\overline{\psi }\,{\vee }\,\overline{\chi }$ and similarly for

  $$\begin{aligned} {\wedge }, {\rightarrow }, \leftrightarrow , \end{aligned}$$

• if $\varphi$ is $(\forall v_{i})\psi$ then $\overline{\varphi }$ is $\forall v_{i}((0\le v_{i}){\wedge } (v_{i}\le v){\rightarrow }\overline{\psi })$,

• if $\varphi$ is $\exists v_{i}\psi$ then $\overline{\varphi }$ is $\exists v_{i}((0\le v_{i}){\wedge } (v_{i}\le v){\rightarrow }\overline{\psi })$.

Thus to any formula $\varphi (v_{1},\ldots , v_{k})$ of ${\mathcal L}_\mathrm{RMV}$ we associate a formula $\overline{\varphi }(v_{1},\ldots , v_{k},v)$ of ${\mathcal L}_\mathrm{Riesz}$. As a consequence, to any sentence $\sigma$ of ${\mathcal L}_\mathrm{RMV}$ corresponds a formula with only one free variable $\overline{\sigma }(v)$ of ${\mathcal L}_\mathrm{Riesz}$.

Proposition 4

Let $(V,u)$ be a Riesz space with strong unit and $R={{\mathrm{\Gamma _{\mathbb R}}}}(V,u)$. If $\sigma$ is a sentence in the first-order theory of Riesz MV-algebras then

$$\begin{aligned} R\models \sigma \,\, \textit{if}\,\, and\,\, only\,\, \textit{if}\,\, V\models \overline{\sigma }[u]. \end{aligned}$$

Proof

By structural induction on terms it follows that $t[a_{1},\ldots , a_{n}]=\overline{t}[a_{1},\ldots , a_{n},u]$ whenever $t(v_{1},\ldots ,v_{n})$ is a term of ${\mathcal L}_{RMV}$ and $a_{1}$, $\ldots$, $a_{n}\in R$. The rest of the proof is straightforward. $\square$

Theorem 4

An equation $\sigma$ in the theory of Riesz MV-algebras holds in all Riesz MV-algebras if and only if it holds in the standard Riesz MV-algebra $[0,1]$.

Proof

One implication is obvious. To prove the other one, let $R$ be a Riesz MV-algebra such that $R\not \models \sigma$. Since $R\simeq {{\mathrm{\Gamma _{\mathbb R}}}}(V,u)$ for some Riesz space with strong unit $(V,u)$, we have that ${{\mathrm{\Gamma _{\mathbb R}}}}(V,u)\not \models \sigma$. Using Proposition 4, we infer that $V\not \models \overline{\sigma }[u]$ in the theory of Riesz spaces. Since the order relation in any lattice can be expressed equationally, we note that $\overline{\sigma }(v)$ is a quasi-identity. By ((Labuschagne and van Alten 2007, Corollary 2.6)) a quasi-identity is satisfied by all Riesz spaces if and only if it is satisfied by $\mathbb {R}$. Hence there exists a real number $c\ge 0$ such that $\mathbb {R}\not \models \overline{\sigma }[c]$. Since $\mathbb {R}\models \overline{\sigma }[0]$, we get $c>0$. If follows that $f:\mathbb {R}\rightarrow \mathbb {R}$ defined by $f(x)\mapsto x/c$ is an automorphism of Riesz spaces. We infer that $\mathbb {R}\not \models \overline{\sigma }[1]$, so $[0,1]\not \models \sigma$. $\square$

Remark 7

The variety of Riesz MV-algebras is generated by the standard model $[0,1]$ in the language of MV-algebras enriched with unary operations $x\mapsto rx$ for any $r\in [0,1]$. This features are reflected by the propositional calculus ${\mathbb R}\mathcal{L}$ presented in Sect. 6, whose Lindenbaum–Tarski algebra is a Riesz MV-algebra.

Since the class of Riesz MV-algebras is a variety, free structures exist. The free Riesz MV-algebra with $n$ free generators is characterized in Corollary 7.

5 Norm-complete Riesz MV-algebras

Let $(V,u)$ be a unital Riesz space and define

$$\begin{aligned}&\Vert \cdot \Vert _u:V\rightarrow {\mathbb R}\,\, \mathrm{by}\\&\Vert x\Vert _u=\inf \{\alpha \ge 0\mid |x|\le \alpha u\}\,\, \mathrm{for\, any}\,\, x\in V. \end{aligned}$$

Then $\Vert \cdot \Vert _u$ is a seminorm ((Meyer-Nieberg 1991, Proposition 1.2.13)) and

$$\begin{aligned} |x|\le |y|\,\, \mathrm{implies}\,\, \Vert x\Vert _u\le \Vert y\Vert _u\,\, \mathrm{for\, any}\,\, x, y\in V. \end{aligned}$$

Remark 8

If $(V,u)$ is a unital Riesz space, then

$$\begin{aligned} \Vert x\Vert _u=\inf \{\alpha \in [0,1]\mid x\le \alpha u\}\,\, \mathrm{for\, any}\,\, x\in [0,u]. \end{aligned}$$

This fact leads us to the following definition, which was suggested by V. Marra (private communication).

Definition 4

(Marra 2009) If $R$ is a Riesz MV-algebra then the unit seminorm $\Vert \cdot \Vert :R\rightarrow [0,1]$ is defined by

$$\begin{aligned} \Vert x\Vert =\inf \{r\in [0,1]\mid x\le r1\} \,\,\mathrm{for\, any}\,\, x\in R. \end{aligned}$$

Remark 9

If $R_1$ and $R_2$ are Riesz MV-algebras and $f: R_1\rightarrow R_2$ is a morphism, then $x\le r1$ in $R_1$ implies $f(x)\le r1$ in $R_2$, so $\Vert f(x)\Vert \le \Vert x\Vert$ for any $x\in R_1$. If $f$ is injective then $\Vert f(x)\Vert =\Vert x\Vert$ for any $x\in R_1$.

This fact allows us to infer properties of the unit seminorm in Riesz MV-algebras directly from the properties of the unit seminorm in Riesz spaces.

Lemma 8

In any Riesz MV-algebra $R$, the following properties hold for any $x$, $y\in R$ and $r\in [0,1]$.

• (a) $\Vert 0\Vert =0$, $\Vert 1\Vert =1$,

• (b) $\Vert x\oplus y\Vert \le \Vert x\Vert +\Vert y\Vert$,

• (c) $x\le y$ implies $\Vert x\Vert \le \Vert y\Vert$,

• (d) $\Vert rx\Vert =r\Vert x\Vert$,

• (e) if $(m-1)x\le x^*$ then $\Vert mx\Vert =m\Vert x\Vert$ for any natural number $m\ge 1$.

Proof

By Theorem 3 and Remark 9 we can safely assume that $R$ is $[0,u]_V$ for some unital Riesz space $(V,u)$. Hence (a)–(d) follow from the properties of the unit seminorm in Riesz spaces ((Fremlin 1974, 25H)).

(e) Note that $(m-1)x\le x^*$ implies

$$\begin{aligned} \underbrace{x\oplus \cdots \oplus x}_{m}=\underbrace{x+\cdots + x}_{m}, \end{aligned}$$

where $+$ is the group addition of $V$, so the desired equality is straightforward. $\square$

Example 3

If $X$ is a compact Hausdorff space, then $C(X)_u=\{f:X\rightarrow [0,1]| f \,\, \mathrm{continuous}\}$ is a Riesz MV-algebra, and for any $f\in C(X)_u$, we have

$$\begin{aligned} \Vert f\Vert&= \inf \{r\in [0,1]\mid f(x)\le r \,\,\forall x\in X\}\\&= \sup \{f(x)|x\in X\}=\Vert f\Vert _\infty . \end{aligned}$$

Recall that an M-space is a unital Riesz space $(V,u)$ that is norm-complete with respect to the unit norm.

Example 4

If $X$ is a compact Hausdorff space and

$$\begin{aligned} C(X)=\{f:X\rightarrow {\mathbb R}\mid f \,\, \text{ continuous }\}, \end{aligned}$$

then $(C(X),{\mathbf 1})$ is an M-space, where ${\mathbf 1}$ is the constant function ${\mathbf 1}(x)=1$ for any $x\in X$.

The above example is fundamental, as proved by Kakutani’s representation theorem.

Theorem 5

(Kakutani 1941) For any M-space $(V,u)$ there exists a compact Hausdorff space $X$ such that $(V,u)$ is isomorphic with $(C(X),{\mathbf 1})$.

Let us denote by ${\mathbb M}{\mathbb U}$ the category of M-spaces with unit-preserving morphisms and by ${\mathbb K}{\mathbb H}aus{\mathbb S}p$ the category of compact Hausdorff spaces with continuous maps.

Theorem 6

(Banaschewski 1976; Johnstone 1982) The category ${\mathbb K}{\mathbb H}aus{\mathbb S}p$ is dual to the category ${\mathbb M}{\mathbb U}$.

We characterize in the sequel those Riesz MV-algebras that are, up to isomorphism, unit intervals in M-spaces. Note that, on any Riesz MV-algebra $R$, we can define

$$\begin{aligned} \delta _{\Vert \cdot \Vert }(x,y)=\Vert d(x,y)\Vert \,\,\mathrm{for\, any}\,\, x, y\in R. \end{aligned}$$

By Lemmas 1 and 8 it follows that $\delta _{\Vert \cdot \Vert }$ is a pseudometric on $R$.

Definition 5

We say that a Riesz MV-algebra $R$ is norm-complete if $(R,\delta _{\Vert \cdot \Vert })$ is a complete metric space.

Theorem 7

If $(V,u)$ is a unital Riesz space then the following are equivalent:

• (i) $(V,u)$ is an M-space,

• (ii) ${{\mathrm{\Gamma _{\mathbb R}}}}(V,u)$ is a norm-complete Riesz MV-algebra.

Proof

We denote $R={{\mathrm{\Gamma _{\mathbb R}}}}(V,u)$.

(i)$\Rightarrow$(ii) By Remark 8, $\Vert x\Vert = \Vert x\Vert _{u}$ for any $x\in [0,u]$. In consequence, any Cauchy sequence w.r.t $\Vert \cdot \Vert$ from $R{{\mathrm{\Gamma _{\mathbb R}}}}(V,u)$ is a Cauchy sequence w.r.t $\Vert \cdot \Vert _{u}$ in $(V,u)$ and we use the fact that $(V,u)$ is norm-complete.

(ii)$\Rightarrow$(i) Let $(v_n)_n$ be a Cauchy sequence in $V$ w.r.t. $\Vert \cdot \Vert _{u}$ such that $v_n\ge 0$ for any $n$. It follows that it is bounded, i.e. there is $y\in V$ and $\Vert v_n\Vert _{u}\le \Vert y\Vert _{u}$ for any $n$. We get $v_n\le \Vert v_n\Vert _{u} u\le \Vert y\Vert _{u}u$ for any $n$, so there exists a natural number $k$ such that $v_n\le ku$ for any $n$. By Lemma 2,

$$\begin{aligned}&v_n=v_{n_1}+\cdots +v_{n_k}, \,\,\mathrm{where}\,\, v_{n_i}\\&\quad =((v_n-(i-1)u)\vee 0)\wedge u\in [0,u] \end{aligned}$$

for any $i\in \{1,\ldots ,k\}$. Since $v_{n_i}\in [0,u]$ we get

$$\begin{aligned} \Vert v_{n_i}\Vert _{u}=\Vert v_{n_i}\Vert \,\,\mathrm{for\, any}\,\, n\,\, \mathrm{and}\,\, i\in \{1,\ldots ,k\}. \end{aligned}$$

One can easily see that $(v_{n_i})_n$ is a Cauchy sequence in $R=[0,u]$ for any $i\in \{1,\ldots ,k\}$. Since $R$ is norm-complete it follows that, for any $i\in \{1,\ldots ,k\}$ there is $w_i\in R$ such that $\lim _n d(v_{n_i},w_i)=\lim _n\Vert v_{n_i}-w_i\Vert _u=0$. If $w=w_1+\cdots +w_k$ then

$$\begin{aligned} \Vert v_n-w\Vert _u\le \Vert v_{n_1}-w_1\Vert _u+\cdots +\Vert v_{n_k}-w_k\Vert _u \end{aligned}$$

for any $n$, so $\lim _n\Vert v_n-w\Vert =0$ and $(v_n)_n$ is convergent w.r.t. $\Vert \cdot \Vert _u$ in $V$.

Recall that $v=(v\vee 0) -((-v)\vee 0)$ for any $v\in V$, so the convergence of arbitrary Cauchy sequences reduces to the convergence of positive Cauchy sequences. $\square$

Denote by ${\mathbb U}{\mathbb R}{\mathbb M}{\mathbb V}$ the category of norm-complete Riesz MV-algebras, which is a full subcategory of ${\mathbb R}{\mathbb M}{\mathbb V}$. By Remark 9, the norm-preserving morphisms coincide with the monomorphisms of ${\mathbb U}{\mathbb R}{\mathbb M}{\mathbb V}$.

Using Theorem 7, the functor ${{\mathrm{\Gamma _{\mathbb R}}}}$ yields the following categorical equivalence.

Corollary 4

The categories ${\mathbb U}{\mathbb R}{\mathbb M}{\mathbb V}$ and ${\mathbb M}{\mathbb U}$ are equivalent.

Corollary 5

The categories ${\mathbb U}{\mathbb R}{\mathbb M}{\mathbb V}$ and ${\mathbb K}{\mathbb H}aus{\mathbb S}p$ are dually equivalent.

Remark 10

Following (Johnstone (1982), Chapter IV) and Banaschewski (1976), the functors establishing the above equivalences are defined on objects as follows:

$$\begin{aligned} R\mapsto \mathrm{Max}(R)\,\, \mathrm{and}\,\, X\mapsto C(X)_u \end{aligned}$$

for any norm-complete Riesz space $R$ and compact Hausdorff space $X$, where Max$(R)$ is the set of all maximal ideals of $R$.

In the sequel, we connect our result with the Gelfand–Naimark duality for C$^*$-algebras (Gelfand and Neimark 1943). Recall that MV-algebras are related with AF C$^*$-algebras in Mundici (1986), but in this case the $K$-theory is used.

Denote ${\mathbb C}^*$ the category whose objects are commutative unital C$^*$-algebras and whose morphisms are unital C$^*$-algebra morphisms.

Theorem 8

((Khalkhali 2009, Chapter 1.1)) The categories ${\mathbb C}^*$ and ${\mathbb K}{\mathbb H}aus{\mathbb S}p$ are dually equivalent.

As a corollary we infer immediately that the categories of commutative unital C$^*$-algebras and norm-complete Riesz MV-algebras are equivalent.

Corollary 6

The categories ${\mathbb C}^*$ and ${\mathbb U}{\mathbb R}{\mathbb M}{\mathbb V}$ are equivalent.

6 The propositional calculus ${\mathbb R}{\mathcal L}$

We denote by ${\mathcal L}_\infty$ the $\infty$-valued propositional Łukasiewicz logic. Recall that ${\mathcal L}_\infty$ has $\lnot$ (unary) and $\rightarrow$ (binary) as primitive connectives, and for any $\varphi$ and $\psi$ we have:

$$\begin{aligned}&\varphi \vee \psi :=(\varphi \rightarrow \psi )\rightarrow \psi , \varphi \wedge \psi :=\lnot (\lnot \varphi \vee \lnot \psi ),\\&\varphi \leftrightarrow \psi :=(\varphi \rightarrow \psi )\wedge (\psi \rightarrow \varphi ).\\ \end{aligned}$$

The language of ${\mathbb R}{\mathcal L}$ contains the language of ${\mathcal L}_\infty$ and a family of unary connectives $\{\nabla _r| r\in [0,1]\}$. We denote by Form$({\mathbb R}{\mathcal L})$ the set of formulas, which are defined inductively as usual.

Definition 6

An axiom of ${\mathbb R}{\mathcal L}$ is any formula that is an axiom of ${\mathcal L}_\infty$ and any formula that has one of the following forms, where $\varphi$, $\psi$, $\chi \in Form({\mathbb R}{\mathcal L})$ and $r$, $q\in [0,1]$:

• (RL1) $\nabla _{r}(\varphi \rightarrow \psi )\leftrightarrow (\nabla _{r}\varphi \rightarrow \nabla _{r}\psi )$;

• (RL2) $\nabla _{(r\odot q^{*})}\varphi \leftrightarrow (\nabla _{q}\varphi \rightarrow \nabla _{r}\varphi )$;

• (RL3) $\nabla _{r}\nabla _{q}\varphi \leftrightarrow \nabla _{(rq)}\varphi$;

• (RL4) $\nabla _{1}\varphi \leftrightarrow \varphi$,

The deduction rule of ${\mathbb R}{\mathcal L}$ is modus ponens and provability is defined as usual.

Remark 11

$\{\varphi \}\vdash \nabla _r\varphi$ is a derived deduction rule for any $r\in [0,1]$.

We recall the usual construction of the Lindenbaum–Tarski algebra. The equivalence relation $\equiv$ is defined on Form$({\mathbb R}{\mathcal L})$ as follows:

$$\begin{aligned} \varphi \equiv \psi \,\,\mathrm{iff}\,\, \vdash \varphi \rightarrow \psi \,\,\mathrm{and}\,\, \vdash \psi \rightarrow \varphi . \end{aligned}$$

We denote by $[\varphi ]$ the equivalence class of a formula $\varphi$ and we define on Form$({\mathbb R}{\mathcal L})/{\equiv }$ the following operations:

$$\begin{aligned}&{[\varphi ]^{*}}= [\lnot \varphi ],\\&\quad {[\varphi ]\oplus [\psi ]}=[\lnot \varphi \rightarrow \psi ], [\varphi ]\odot [\psi ]=[\lnot (\varphi \rightarrow \lnot \psi )],\\&\quad 0=[\lnot (v_1\rightarrow v_1)], 1=0^*=[v_1\rightarrow v_1]. \end{aligned}$$

To define the scalar multiplication we introduce new connectives:

$$\begin{aligned} \Delta _r\varphi :=\lnot (\nabla _r\lnot \varphi ) \end{aligned}$$

and we set $r [\varphi ]= [\Delta _r\varphi ]$ for any $r\in [0,1]$ and $\varphi$ formula of ${\mathbb R}{\mathcal L}$.

Proposition 5

The Lindenbaum–Tarski algebra

$$\begin{aligned} RL=(Form({\mathbb R}{\mathcal L})/{\equiv },\cdot , \oplus ,^{*}, 0) \end{aligned}$$

is a Riesz MV-algebra.

Proof

The axioms (RL1)–(RL4) are logical expressions of the duals of (RMV1)–(RMV4). We prove in detail that $RL$ satisfies (RMV1). If $\varphi$ and $\psi$ are two formulas and $r\in [0,1]$ then, by (RL1), we get

$$\begin{aligned}{}[\nabla _r(\lnot \varphi \rightarrow \lnot \psi )]=[\nabla _r\lnot \varphi \rightarrow \nabla _r\lnot \psi ]. \end{aligned}$$

It follows that:

$$\begin{aligned}&[\nabla _r\lnot (\lnot \varphi \odot \psi )]=[\lnot (\nabla _r\lnot \varphi \odot \lnot \nabla _r\lnot \psi )]\\&\quad [\lnot \nabla _r\lnot (\lnot \varphi \odot \psi )]=[\nabla _r\lnot \varphi \odot \lnot \nabla _r\lnot \psi ]\\&\quad [\Delta _r(\lnot \varphi \odot \psi )]=[\lnot \Delta _r\varphi \odot \nabla _r\psi ]\\&\quad r([\varphi ]^*\odot [\psi ])=(r[\varphi ])^*\odot (r[\psi ]), \end{aligned}$$

so (RMV1) holds in $RL$. $\square$

Let $R$ be an Riesz MV-algebra. An evaluation is a function $e:Form({\mathbb R}{\mathcal L})\rightarrow R$ which satisfies the following conditions for any $\varphi$, $\psi \in Form({\mathbb R}{\mathcal L})$ and $r\in [0,1]$:

• (e1) $e(\varphi \rightarrow \psi )=e(\varphi )^*\oplus e(\psi )$,

• (e2) $e(\lnot \varphi )=e(\varphi )^{*}$,

• (e3) $e(\nabla _{r}\varphi )=(re(\varphi )^*)^{*}$.

As a consequence of Theorem 4, the propositional calculus ${\mathbb R}\mathcal{L}$ is complete with respect to $[0,1]$.

Theorem 9

For a formula $\varphi$ of ${\mathbb R}\mathcal{L}$ the following are equivalent:

• (i) $\varphi$ is provable in ${\mathbb R}\mathcal{L}$,

• (ii) $e(\varphi )=1$ for any Riesz MV-algebra $R$ and for any evaluation $e:Form({\mathbb R}{\mathcal L})\rightarrow R$,

• (iii) $e(\varphi )=1$ for any evaluation $e:Form({\mathbb R}{\mathcal L})\rightarrow [0,1]$.

Remark 12

The system ${\mathbb R}{\mathcal L}$ is a conservative extension of ${\mathcal L}_\infty$, i.e. a formula $\varphi$ of ${\mathcal L}_\infty$ is a theorem of ${\mathcal L}_\infty$ if and only if it is a theorem of ${\mathbb R}{\mathcal L}$ . Since any proof in ${\mathcal L}_\infty$ is also a proof in ${\mathbb R}{\mathcal L}$, one implication is obvious. To prove the other one, assume that $\varphi$ is a formula of ${\mathcal L}_\infty$ which is not a theorem of ${\mathcal L}_\infty$. Hence there exists an evaluation $e^\prime : \mathrm{Form}({\mathcal L}_\infty )\rightarrow [0,1]$ such that $e^\prime (\varphi )\ne 1$. Let $e:\mathrm{Form}({\mathbb R}{\mathcal L})\rightarrow [0,1]$ be the unique evaluation in ${\mathbb R}{\mathcal L}$ such that $e(v)=e^\prime (v)$ for any propositional variable $v$. By structural induction on formulas one can prove that $e(\psi )=e^\prime (\psi )$ for any $\psi \in \mathrm{Form}({\mathcal L}_\infty )$. It follows that $e(\varphi )=e^\prime (\varphi )\ne 1$, so $\varphi$ is not a theorem of ${\mathbb R}{\mathcal L}$.

Remark 13

A formula $\varphi$ with variables from $\{v_1,\ldots ,v_n\}$ defines a term function:

$$\begin{aligned} \widetilde{\varphi }:[0,1]^n\rightarrow [0,1], \widetilde{\varphi }(x_1,\ldots ,x_n)=e(\varphi ), \end{aligned}$$

where $e$ is an evaluation such that $e(v_i)=x_i$ for any $i\in \{1,\ldots ,n\}$. By Theorem 9 it follows that $[\varphi ]=[\psi ]$ if and only if $\widetilde{\varphi } = \widetilde{\psi }$.

7 Term functions and piecewise linear functions

In the following, we characterize the class of functions that can be defined by formulas in ${\mathbb R}{\mathcal L}$.

Definition 7

Let $n>1$ be a natural number. A piecewise linear function is a function $f:{\mathbb R}^n\rightarrow {\mathbb R}$ for which there exists a finite number of affine functions

$$\begin{aligned} q_1, \ldots , q_k:\mathbb {R}^n\rightarrow \mathbb {R}\end{aligned}$$

and for any $(x_1,\ldots , x_n)\in {\mathbb R}^n$ there is $i\in \{1,\ldots , k\}$ such that $f(x_1,\ldots , x_n)=q_i(x_1,\ldots , x_n)$. We say that $q_1$, $\ldots$, $q_k$ are the components of $f$.

We denote by $PL_n$ the set of all continuous functions $f:[0,1]^n\rightarrow [0,1]$ that are piecewise linear.

For the rest of the paper, all piecewise linear functions are continuous.

Theorem 10

If $\varphi$ is a formula of ${\mathbb R}{\mathcal L}$ with propositional variables from $\{v_1,\ldots ,v_n\}$ then $\widetilde{\varphi }\in PL_n$.

Proof

We prove the result by structural induction on formulas.

If $\varphi$ is $v_i$ for some $i\in \{1,\ldots ,n\}$ then $\widetilde{\varphi }=\pi _i$ (the $i$-th projection).

If $\varphi$ is $\lnot \psi$ and $q_{1}$, $\ldots$, $q_{s}$ are the components of $\widetilde{\psi }$, then $1-q_{1}$, $\ldots$, $1-q_{s}$ are the components of $\widetilde{\varphi }$.

Assume $\varphi$ is $\psi \rightarrow \chi$. If $q_{1}$, $\ldots$, $q_{m}$ are the components of $\widetilde{\psi }$ and $p_{1}$, $\ldots$, $p_{k}$ are the components of $\widetilde{\chi }$, then $\widetilde{\varphi }$ is defined by $\{1\}\cup \{s_{ij}\}_{i,j}$, where $s_{ij}=1-q_i+p_j$ for any $i\in \{1,\ldots ,s\}$ and $j\in \{1,\ldots ,r\}$.

If $\varphi$ is $\Delta _r\psi$ for some $r\in [0,1]$ and $q_{1}$, $\ldots$, $q_{s}$ are the components of $\widetilde{\psi }$, then $1-r+rq_{1}$, $\ldots$, $1-r+rq_{s}$ are the components of $\widetilde{\varphi }$. $\square$

Remark 14

The continuous piecewise linear functions $f:[0,1]^n\rightarrow [0,1]$ with integer coefficients are called McNaughton functions and they are in one–one correspondence with the formulas of Łukasiewicz logic by McNaughton (1951) theorem. The continuous piecewise linear functions with rational coefficients correspond to formulas of Rational Łukasiewicz logic, a propositional calculus developed in Gerla (2001) that has divisible MV-algebras as models. In Theorem 11 we prove that any continuous piecewise linear function with real coefficients $f:[0,1]^n\rightarrow [0,1]$ is the term function of a formula from ${\mathbb R}{\mathcal L}$.

For now on we define $\varrho :{\mathbb R}\rightarrow [0,1]$ by

$$\begin{aligned} \varrho (x)=(x\vee 0)\wedge 1 \,\,\mathrm{for\, any}\,\, x\in {\mathbb R}. \end{aligned}$$

Lemma 9

For any $x$, $y\in {\mathbb R}$ the following holds:

• (a) $(x\vee 0) +(y\vee 0)\ge (x+y)\vee 0$,

• (b) $x\ge 0$ iff $\varrho (-x)=0$,

• (c) $\varrho (x)=\varrho (x\vee 0)$.

Proof

• (a) $(x\vee 0) +(y\vee 0)=(x+y)\vee x\vee y\vee 0\ge (x+y)\vee 0$.

• (b) $\varrho (-x)=0$ iff $((-x)\vee 0)\wedge 1=0$ iff $(-x)\vee 0=0$ iff $-x\le 0$ iff $x\ge 0$.

• (c) $\varrho (x\vee 0)=(x\vee 0\vee 0)\wedge 1=(x\vee 0)\wedge 1 =\varrho (x)$.

$\square$

The next lemma generalizes some results from (Cignoli et al. (2000), Lemma 3.1.9).

Lemma 10

If $g:[0,1]^n\rightarrow {\mathbb R}$ and $h:[0,1]^n\rightarrow [0,1]$ then the following properties hold:

• (a) $\varrho \circ (g+h) = ((\varrho \circ g)\oplus h)\odot (\varrho \circ (g+1))$.

• (b) $\varrho \circ (1-g)=1-(\varrho \circ g)$

Proof

Let $\mathbf {x}=(x_1,\ldots , x_n)$ be an element from $[0,1]^n$.

(a) If $g(\mathbf {x})>1$ then $g(\mathbf {x})+1>1$ and $g(\mathbf {x})+h(\mathbf {x})>1$. It follows that

$$\begin{aligned} \varrho ({g}(\mathbf {x}))=\varrho ({(g+1)}(\mathbf {x}))=\varrho ((g+h)(\mathbf {x}))=1, \end{aligned}$$

so the intended identity is obvious.

If $g(\mathbf {x})\in [0,1]$ then $\varrho ({g}(\mathbf {x}))=g(\mathbf {x})$ and $\varrho ({(g+1)}(\mathbf {x}))=1$ for any $\mathbf {x}\in [0,1]^n$, so

$$\begin{aligned} \varrho ({(g+h)}(\mathbf {x}))&= h(\mathbf {x})\oplus g(\mathbf {x})\\&= h(\mathbf {x})\oplus \varrho ({g}(\mathbf {x}))=((\varrho \circ {g})\oplus h)(\mathbf {x})\\&= (((\varrho \circ {g})\oplus h)\odot 1)(\mathbf {x})\\&= (((\varrho \circ {g})\oplus h)\odot (\varrho \circ {(g+1)}))(\mathbf {x}). \end{aligned}$$

Assume that $g(\mathbf {x})<0$, so $\varrho ({g}(\mathbf {x}))=0$. We have to prove that $\varrho ({(g+h)}(\mathbf {x}))=h(\mathbf {x})\odot \varrho ({(g+1)}(\mathbf {x}))$.

If $g(\mathbf {x})\le -1$ then $\varrho (g(\mathbf {x}))=\varrho ({(g+1)}(\mathbf {x}))=0$ and $g(\mathbf {x})+h(\mathbf {x})\le -1+h(\mathbf {x})\le 0$, so

$$\begin{aligned} \varrho ({(g+h)}(\mathbf {x}))=0=h(\mathbf {x})\odot 0=(h\odot (\varrho \circ {(g+1)}))(\mathbf {x}). \end{aligned}$$

If $g(\mathbf {x})\in (-1,0)$ then $\varrho ({(g+1)}(\mathbf {x}))=(g+1)(\mathbf {x})=g(\mathbf {x})+1$, so we get

$$\begin{aligned} (\varrho \circ {(g+h)})(\mathbf {x})&= 0\vee (1\wedge (h(\mathbf {x})+g(\mathbf {x})))\\&= 0\vee (h(\mathbf {x})+g(\mathbf {x}))\\&= 0\vee (h(\mathbf {x})+g(\mathbf {x})+1-1)\\&= h(\mathbf {x})\odot (g(\mathbf {x})+1)\\&= h(\mathbf {x})\odot \varrho ({(g+1)}(\mathbf {x}))\\&= (h\odot (\varrho \circ {(g+1)}))(\mathbf {x}). \end{aligned}$$

(b) If $g(\mathbf {x})<0$ then $\varrho ({g}(\mathbf {x}))=0$ and

$$\begin{aligned} \varrho ({(1-g)}(\mathbf {x}))=1=1-0=1-\varrho ({g}(\mathbf {x})). \end{aligned}$$

If $g(\mathbf {x})\in [0,1]$ then $(1-g)(\mathbf {x})\in [0,1]$, so

$$\begin{aligned} \varrho ({(1-g)}(\mathbf {x}))=(1-g)(\mathbf {x}) =1-g(\mathbf {x})=1-\varrho ({g}(\mathbf {x})). \end{aligned}$$

If $g(\mathbf {x})>1$ then $\varrho ({g}(\mathbf {x}))=1$ and

$$\begin{aligned} \varrho ({(1-g)}(\mathbf {x}))=0=1-1=1-\varrho ({g}(\mathbf {x})). \end{aligned}$$

$\square$

Proposition 6

For any affine function $f:[0,1]^n\rightarrow {\mathbb R}$ there exists a formula $\varphi$ of ${\mathbb R}{\mathcal L}$ such that $\varrho \circ f=\widetilde{\varphi }$.

Proof

Let $f:[0,1]^n\rightarrow {\mathbb R}$ be an affine function, i.e. there are $c_0$, $\ldots$, $c_n\in {\mathbb R}$ such that

$$\begin{aligned} f(x_1,\ldots , x_n)=c_nx_n+\cdots +c_1x_1+c_0 \end{aligned}$$

for any $(x_1,\ldots , x_n)\in [0,1]^n$. Note that for any $c\in {\mathbb R}$ there is a natural number $m$ such that $c=r_1+\cdots +r_m$ where $r_1$, $\ldots$, $r_m\in [-1,1]$. Hence we assume that

$$\begin{aligned} f(x_1,\ldots , x_n)=r_my_m+\cdots + r_{p+1}y_{p+1}+r_{p}+\ldots +r_1 \end{aligned}$$

where $m\ge 1$ and $0\le p\le m$ are natural numbers, $r_j\in [-1,1]\setminus \{0\}$ for any $j\in \{1,\ldots , m\}$ and

$y_j\in \{x_1,\ldots ,x_n\}$ for any $j\in \{p+1,\ldots , m\}$.

We prove the theorem by induction on $m\ge 1$. Let us denote $\mathbf {x}=(x_1,\ldots , x_n)$ an element from $[0,1]^n$

Initial step $m=1$. We have $f(\mathbf {x})=r$ for any $\mathbf {x}\in [0,1]^n$ or $f(\mathbf {x})=rx_i$ for any $\mathbf {x}\in [0,1]^n$, where $r\in [-1,1]\setminus \{0\}$ and $i\in \{1,\ldots ,n\}$. If $r\in [-1,0)$ then $\varrho \circ f=0$ so $\varrho \circ f=\widetilde{\varphi }$ for $\varphi =v_1\odot \lnot v_1$ . If $r\in (0,1]$ then $f=\varrho \circ f$. It follows that $f=\widetilde{\varphi }$ where $\varphi =\nabla _r(v_1\rightarrow v_1)$ if $f(\mathbf {x})=r$ for any $\mathbf {x}\in [0,1]^n$ and $\varphi ={\nabla _r v_i}$ if $f(\mathbf {x})=rx_i$ for any $\mathbf {x}\in [0,1]^n$.

Induction step We take $f=g+h$ where $\varrho \circ g=\widetilde{\varphi }$ for some formula $\varphi$ and there are $r\in [-1,1]\setminus \{0\}$ and $i\in \{1,\ldots ,n\}$ such that $h(\mathbf {x})=r$ for any $\mathbf {x}\in [0,1]^n$, or $h(\mathbf {x})=rx_i$ for any $\mathbf {x}\in [0,1]^n$. We consider two cases.

Case 1. If $r\in (0,1]$ then $h:[0,1]^n\rightarrow [0,1]$ so

$$\begin{aligned} \varrho \circ f =((\varrho \circ g)\oplus h)\odot (\varrho \circ (1+g)) \end{aligned}$$

by Lemma 10 (a). Following the initial step, there is a formula $\psi$ such that $h=\widetilde{\psi }$. Note that $1+g=1-(-g)$ and, since the induction hypothesis holds for $(-g)$, there is a formula $\chi$ such that $\varrho \circ (-g)=\widetilde{\chi }$. In consequence, by Lemma 10 (b), $\varrho \circ (1+g)=1-\widetilde{\chi }=\widetilde{\lnot \chi }$. We get $\varrho \circ f=\widetilde{\theta }$ where $\theta ={(\varphi \oplus \psi )\odot \lnot \chi }$.

Case 2. If $r\in [-1,0)$, then $g+h=(g-1)+(1+h)$ and $1+h:[0,1]^n\rightarrow [0,1]$. By Lemma 10 (a) we get

$$\begin{aligned} \varrho \circ f =((\varrho \circ (g-1))\oplus (1+h))\odot (\varrho \circ g). \end{aligned}$$

Following the initial step, there is a formula $\psi$ such that

$$\begin{aligned} -h=\widetilde{\psi }, \,\,\mathrm{so}\,\, 1+h=1-(-h)=\widetilde{\lnot \psi }. \end{aligned}$$

In the sequel we have to find a formula $\chi$ that corresponds to $\varrho \circ (g-1)$, where

$$\begin{aligned} g(\mathbf {x})=r_my_m+\cdots r_{p+1}y_{p+1}+r_{p}+\cdots +r_1 \end{aligned}$$

with $r_j\in [-1,1]\setminus \{0\}$ for any $j\in \{1,\cdots , m\}$ and $y_j\in \{x_1,\ldots ,x_n\}$ for any $j\in \{p+1,\ldots , m\}$.

Case 2.1 If $r_j\le 0$ for any $j\in \{1,\ldots , m\}$ then $g-1\le 0$, so $\varrho \circ (g-1)=0= \widetilde{\chi }$ with $\chi =v_1\odot \lnot v_1$.

Case 2.2 If there is $j_0\in \{1,\ldots , p\}$ such that $r_{j_0}>0$, then it follows that

$$\begin{aligned} (g-1)(\mathbf {x})&= r_my_m+\cdots + r_{p+1}y_{p+1}+r_{p}+\ldots +(r_{j_0}-1)\\&+\cdots +r_1 \end{aligned}$$

and $r_{j_0}-1\in [-1,0)$, so the induction hypothesis applies to $g-1$. In consequence, there exists a formula $\chi$ such that $\varrho \circ (g-1)=\widetilde{\chi }$.

Case 2.3 If there is $j_0\in \{p+1,\ldots , m\}$ such that $r_{j_0}>0$, then we set $h_0(\mathbf {x})=r_{j_0}y_{j_0}$ and

$$\begin{aligned} g_0(\mathbf {x})=g(\mathbf {x})-r_{j_0}y_{j_0}-1. \end{aligned}$$

It follows that $g-1=g_0+h_0$ such that $g_0$ satisfies the induction hypothesis and $h_0:[0,1]^n\rightarrow [0,1]$. We are in the hypothesis of Case 1, so there exists a formula $\chi$ such that $\varrho \circ (g-1)=\widetilde{\chi }$.

Summing up, we get $\varrho \circ (g+h) =\widetilde{\theta }$ with $\theta ={((\chi \oplus \lnot \psi )\odot \varphi )}$. $\square$

Theorem 11

For any $f:[0,1]^n\rightarrow [0,1]$ from $PL_n$ there is a formula $\varphi$ of ${\mathbb R}{\mathcal L}$ such that $f=\widetilde{\varphi }$.

Proof

Let $f:[0,1]^n\rightarrow [0,1]$ be in $PL_n$. Using the Max-Min representation from Ovchinnikov (2002), there are finite sets $I$ and $J$ such that

$$\begin{aligned} f=\bigvee _{i\in I}\bigwedge _{j\in J} f_{ij}, \end{aligned}$$

where $f_{ij}:[0,1]^n\rightarrow {\mathbb R}$ are affine functions. We note that

$$\begin{aligned} f=\varrho \circ f=\bigvee _{i\in I}\bigwedge _{j\in J}( \varrho \circ f_{ij}). \end{aligned}$$

By Proposition 6, for any $i\in I$ and $j\in J$ there is a formula $\varphi _{ij}$ such that $\varrho \circ f_{ij}=\widetilde{\varphi _{ij}}$. In consequence, if we set $\varphi =\bigvee _{i\in I}\bigwedge _{j\in J} \varphi _{ij}$ then $f=\widetilde{\varphi }$. $\square$

For any $n\ge 1$, the set $PL_n$ is a Riesz MV-algebra with the operations defined componentwise. If $RL_n$ is the Lindenbaum–Tarski algebra of ${\mathbb R}{\mathcal L}$ defined on formulas with variables from $\{v_1,\ldots , v_n\}$, then $RL_n$ is the free Riesz MV-algebra with $n$ free generators by standard results in universal algebra Burris and Sankappanavar (1982). Since the function $[\varphi ]\mapsto \widetilde{\varphi }$ is obviously an isomorphism between $RL_n$ and $PL_n$ the following corollary is straightforward.

Corollary 7

$PL_n$ is the free Riesz MV-algebra with $n$ free generators.

8 Linear combinations of formulas and de Finetti’s coherence criterion

We recall in the beginning de Finetti’s coherence criterion for boolean events. If $S=\{\varphi _1,\ldots , \varphi _k\}$ is a set of classical events then a book is a set

$$\begin{aligned} \{(\varphi _i,r_i)\mid i\in \{1,\ldots , k\}\}, \end{aligned}$$

where $r_i\in [0,1]$ is a “betting odd” assigned by a bookmaker for $\varphi _i$ for any $i\in \{1,\ldots , k\}$. The book is coherent if there is no system of bets $\{c_1,\ldots , c_k\}$ which causes the bookmaker a sure loss. This means that for any real numbers $\{c_1,\ldots , c_k\}$ there exists an evaluation $e:S\rightarrow \{0,1\}$ such that $\sum _{i=1}^kc_i(r_i-e(\varphi _i))\ge 0$. De Finetti’s coherence criterion de Finetti (1931) states that the book $\{(\varphi _i,r_i)|i\in \{1,\ldots , k\}\}$ is coherent if there is a boolean probability $\mu$ defined on the algebra of events generated by $S$ such that $\mu (\varphi _i)=r_i$ for any $i\in \{1,\ldots ,k\}$. When the underlying logic is Łukasiewicz logic, the events belongs to an MV-algebra and they are evaluated in $[0,1]$ (Mundici 2006). Consequently, the coherence criterion uses states instead of boolean probabilities.

We recall the MV-algebraic approach to de Finetti’s notion of coherence from Kühr and Mundici (2007) and prove a similar coherence criterion for Riesz MV-algebras. We also provide a logical expression of the coherence criterion when the events are represented by formulas ${\mathbb R}{\mathcal L}$. To accomplish this task, we initiate the study of linear combinations of formulas in ${\mathbb R}{\mathcal L}$.

The next result characterizes the states defined on Riesz MV-algebras.

Lemma 11

If $R$ is a Riesz MV-algebra then any state $s: U(R)\rightarrow [0,1]$ is also homogeneous:

$$\begin{aligned} (s3)\, s(r\cdot x)=rs(x)\,\, for\,\, any\,\, r\,\in [0,1], x\in R. \end{aligned}$$

Proof

We can safely assume that $R=[0,u]_V$ for some unital Riesz space $(V,u)$. By ((Mundici 1995, Theorem 2.4)), there is a state $s^\prime :V\rightarrow {\mathbb R}$ such that $s^\prime (x)=s(x)$ for any $x\in [0,u]$.

If $r\in [0,1]\cap {\mathbb Q}$ then $r=\frac{m}{n}$ and $rx=y$ in $[0,u]$ implies $mx=ny$ in $V$. It follows that $s^\prime (mx)=s^\prime (ny)$, so $ms(x)=ns(y)$ and we get $s(rx)=s(y)=rs(x)$.

If $r\in (0,1)$ there are rational sequences $(r_n)_n$ and $(q_n)_n$ such that $r_n\uparrow r$ and $q_n\downarrow r$. Hence

$$\begin{aligned} r_ns(x)=s(r_nx)\le s(rx)\le s(q_nx)=q_ns(x) \end{aligned}$$

for any $n$ and $x\in [0,1]$. The intended result follows by an application of Stolz–Cesàro theorem. $\square$

Definition 8

If $R$ is a Riesz MV-algebra, a state on $R$ is a function $s:R\rightarrow [0,1]$ which satisfies the conditions (s1), (s2) and (s3) (additivity, normalization and homogeneity).

The previous lemma asserts that the states of a Riesz MV-algebra $R$ coincide with the states of its MV-algebra reduct ${{\mathrm{U}}}(R)$.

The following definition generalizes de Finetti’s notion of coherence and provides an algebraic approach within Łukasiewicz logic.

Definition 9

(Kühr and Mundici 2007) If $A$ is an MV-algebra and $x_1, \ldots , x_k$ are in $A$ then a map $\beta :\{x_1,\ldots , x_n\} \!\rightarrow \! [0,1]$ is coherent if for any $c_1$, $\ldots$, $c_k\in {\mathbb R}$ there exists a morphism of MV-algebras $e:A\rightarrow [0,1]$ such that

$$\begin{aligned} \sum _{i=1}^kc_i(\beta (x_i)-e(x_i))\ge 0. \end{aligned}$$

Theorem 12

((Kühr and Mundici 2007, Theorem 3.2)) If $A$ is an MV-algebra, $x_1$, $\ldots$, $x_k\in A$ and $\beta :\{x_1,\ldots , x_n\} \rightarrow [0,1]$ then the following are equivalent:

• (i) the map $\beta$ is coherent,

• (ii) there exists a state $s: A\rightarrow [0,1]$ such that

  $$\begin{aligned} s(x_i)=\beta (x_i)\,\, for\,\, any\,\, i\, \in \{1,\ldots ,k\}, \end{aligned}$$

• (iii) there exists $e_1$, $\ldots$, $e_m:A\rightarrow [0,1]$ morphisms of MV-algebras such that $m\le k + 1$ and $\beta$ is the restriction of a convex combination of $\{e_1, \ldots , e_m\}$.

By Remark 4, any morphsim of Riesz MV-algebras is just a morphism between the MV-algebra reducts of its domain and codomain. Therefore, the notion of coherent map remains unchanged on Riesz MV-algebras and an analogue of Theorem 12 can be proved for Riesz MV-algebras as well.

Corollary 8

If $R$ is a Riesz MV-algebra, $x_1$, $\ldots$, $x_k\in R$ and $\beta :\{x_1,\ldots , x_k\} \rightarrow [0,1]$ then the following are equivalent:

• (i) the map $\beta$ is coherent,

• (ii) there exists a state $s: R\rightarrow [0,1]$ such that

  $$\begin{aligned} s(x_i)=\beta (x_i)\,\, for\,\, any\,\, i\, \in \{1,\ldots ,k\}, \end{aligned}$$

• (iii) there exists $e_1$, $\ldots$, $e_m:R\rightarrow [0,1]$ morphisms such that $m\le k + 1$ and $\beta$ is the restriction of a convex combination of $\{e_1, \ldots , e_m\}$.

Proof

(i) $\Leftrightarrow$ (iii) and (ii) $\Rightarrow$ (i) follow by Theorem 12 applied to the MV-algebra reduct of $R$ and by Remark 4.

(iii)$\Rightarrow$(ii) There are $\alpha _1$, $\ldots$, $\alpha _m\in [0,1]$ such that

$$\begin{aligned}&\alpha _1+\cdots + \alpha _m=1\,\, \mathrm{and}\\&\beta (x_i)=\alpha _1e_1(x_i)+\cdots + \alpha _me_m(x_i) \end{aligned}$$

for any $i\in \{1,\ldots ,k\}$. We set $s= \alpha _1e_1+\cdots + \alpha _me_m$ that satisfies (s1), (s2) and (s3), so $s:R\rightarrow [0,1]$ is the required state. $\square$

The above result is an algebraic version of de Finetti’s coherence criterion. In the sequel we provide a logical approach within ${\mathbb R}\mathcal{L}$.

We firstly recall de Finetti’s coherence criterion for Łukasiewicz logic ${\mathcal L}_\infty$.

Theorem 13

(Mundici 2006) If $\varphi _1, \ldots , \varphi _k$ are formulas of ${\mathcal L}_\infty$ with variables ${v_1,\ldots ,v_n}$ and $r_1, \ldots , r_k\in [0,1]$ then the following are equivalent:

• (i) the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent,

• (ii) there exists a state $s:L_n\rightarrow [0,1]$ such that

$$\begin{aligned} s([\varphi _i])=r_i for any i\in \{1,\ldots ,k\}, \end{aligned}$$

where $L_n$ is the Lindenbaum–Tarski algebra of the formulas in $n$ variables and $[\varphi ]$ is the equivalence class of the formula $\varphi$ in $L_n$.

When we consider ${\mathbb R}{\mathcal L}$ instead of ${\mathcal L}_\infty$, we get the following.

Definition 10

If $\varphi _1,\ldots ,\varphi _k$ are formulas of ${\mathbb R}\mathcal{L}$ and $r_1$, $\ldots$, $r_k\in [0,1]$ then the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent if for any $c_1$, $\ldots$, $c_k\in {\mathbb R}$ there exists an evaluation $e:\mathrm{Form}({\mathbb R}\mathcal{L})\rightarrow [0,1]$ such that

$$\begin{aligned} \sum _{i=1}^kc_i(r_i-e(\varphi _i))\ge 0. \end{aligned}$$

To characterize coherence in logical terms within ${\mathbb R}{\mathcal L}$ we introduce the notion of quasi-linear combination of piecewise linear functions.

In the sequel we assume that $n$ is a natural number and all the formulas from ${\mathbb R}\mathcal{L}$ have variables from the set $\{v_1,\ldots ,v_n\}$. As in the previous section, we use the function

$$\begin{aligned} \varrho :{\mathbb R}\rightarrow [0,1], \varrho (x)=(x\vee 0)\wedge 1\,\, \mathrm{for\, any\, x}\,\, \in {\mathbb R}. \end{aligned}$$

Remark 15

If $f_1$, $\ldots$, $f_k :[0,1]^n\rightarrow {\mathbb R}$ are continuous piecewise linear functions and $c_1$, $\ldots$, $c_k\in {\mathbb R}$ then

$\sum _{i=1}^k c_if_i$ is also a continuous piecewise linear function, so $\varrho \circ (\sum _{i=1}^k c_if_i)$ is in $PL_n$. By Theorem 11, there exists a formula $\varphi$ of ${\mathbb R}\mathcal{L}$ such that $\widetilde{\varphi }=\varrho \circ (\sum _{i=1}^k c_if_i)$. Therefore, we introduce the following definition.

Definition 11

Let $f_1$, $\ldots$, $f_k:[0,1]^n\rightarrow {\mathbb R}$ be continuous piecewise linear functions. We say that $\varphi$ is a quasi-linear combination of $f_1$, $\ldots$, $f_k$ whenever

$$\begin{aligned} \widetilde{\varphi }=\varrho \circ \left( \sum _{i=1}^k c_if_i\right) \end{aligned}$$

for some $c_1, \ldots , c_k\in {\mathbb R}$. We define $qspan(f_1, \ldots , f_k)$ as the subset of $Form({\mathbb R}\mathcal{L})$ that contains all the quasi-linear combinations of $f_1$, $\ldots$, $f_k$. If $\varphi _1$, $\ldots$, $\varphi _k$ are formulas of ${\mathbb R}\mathcal{L}$ then $qspan(\widetilde{\varphi _1}, \ldots , \widetilde{\varphi _k})$ will be denoted by $qspan(\varphi _1, \ldots , \varphi _k)$. If $\varphi \in qspan(\varphi _1, \ldots , \varphi _k)$ then we say that $\varphi$ is a quasi-linear combination of the formulas $\varphi _1$, $\ldots$, $\varphi _k$ in ${\mathbb R}\mathcal{L}$.

Lemma 12

If $\varphi$, $\varphi _1$, $\ldots$, $\varphi _k$ are formulas in ${\mathbb R}\mathcal{L}$ such that $\widetilde{\varphi }=\varrho \circ (\sum _{i=1}^k c_i\widetilde{\varphi _i})$ for some $c_1$, $\ldots$, $c_k\in {\mathbb R}$ and $e:Form({\mathbb R}{\mathcal L})\rightarrow [0,1]$ is an evaluation, then

$$\begin{aligned} e(\varphi )= \varrho \left( \sum _{i=1}^k c_i e(\varphi _i)\right) . \end{aligned}$$

Proof

We have $e(\varphi )=\widetilde{\varphi }(e(v_1),\ldots , e(v_n))=$

$\varrho (\sum _{i=1}^k c_i \widetilde{\varphi _i}(e(v_1),\ldots , e(v_n)))=\varrho (\sum _{i=1}^k c_i e(\varphi _i))$. $\square$

Lemma 13

If $\varphi _1$, $\ldots$, $\varphi _k$ are formulas of ${\mathbb R}{\mathcal L}$ and $r_1$, $\ldots$, $r_k\in [0,1]$ then

$$\begin{aligned} \nabla _{r_1}\varphi _1\oplus \ldots \oplus \nabla _{r_k}\varphi _k \in qspan(\varphi _1, \ldots ,\varphi _k). \end{aligned}$$

Proof

Under the above hypothesis, we get

$$\begin{aligned} \varrho \left( \sum _{i=1}^k r_i e(\varphi _i)\right)&= r_1e(\varphi _1)\oplus \cdots \oplus {r_k}e(\varphi _k)\\&= e(\nabla _{r_1}\varphi _1)\oplus \cdots \oplus e(\nabla _{r_k}\varphi _k) \end{aligned}$$

for any evaluation $e:\mathrm{Form}({\mathbb R}{\mathcal L})\rightarrow [0,1]$. $\square$

Definition 12

We say that a formula $\varphi$ of ${\mathbb R}{\mathcal L}$ is invalid if there exists an evaluation $e:Form({\mathbb R}{\mathcal L})\rightarrow [0,1]$ such that $e(\varphi )=0$

Theorem 14

Let $\varphi _1$, $\ldots$, $\varphi _k$ are formulas of ${\mathbb R}{\mathcal L}$ and $r_1$, $\ldots$, $r_k\in [0,1]$. The following are equivalent:

• (i) the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent,

• (ii) there exists a state $s:RL_n\rightarrow [0,1]$ such that $s([\varphi _i])=r_i$ for any $i\in \{1,\ldots ,k\}$,

• (iii) $qspan(\widetilde{\varphi _1}-r_1,\ldots , \widetilde{\varphi _k}-r_k)$ is a set of invalid formulas of ${\mathbb R}\mathcal{L}$.

Proof

(i)$\Leftrightarrow$(ii) Apply Corollary 8 to $\beta ([\varphi _i])=r_i$ for any $i\in \{1,\ldots ,k\}$.

(i)$\Leftrightarrow$(iii) The following facts are equivalent:

• (1) $qspan(\widetilde{\varphi _1}-r_1,\ldots , \widetilde{\varphi _k}-r_k)$ is a set of invalid formulas,

• (2) for any $\Psi \in qspan(\widetilde{\varphi _1}-r_1,\ldots , \widetilde{\varphi _k}-r_k)$ there exists an evaluation $e:\mathrm{Form}({\mathbb R}{\mathcal L})\rightarrow [0,1]$ such that $e(\Psi )=0$,

• (3) for any $c_1$, ..., $c_k\in {\mathbb R}$, if $\widetilde{\Psi }=\varrho \circ \left( \sum _{i=1}^k c_i(\widetilde{\varphi _i}-r_i)\right)$ then there exists an evaluation $e:\mathrm{Form}({\mathbb R}{\mathcal L})\rightarrow [0,1]$ such that $e(\Psi )=0$,

• (4) for any $c_1$, ..., $c_k\in {\mathbb R}$, there exists an evaluation $e:Form({\mathbb R}{\mathcal L})\rightarrow [0,1]$ such that

  $$\begin{aligned} \varrho \circ \left( \sum _{i=1}^k c_i(e(\varphi _i)-r_i\right) =0, \end{aligned}$$

• (5) for any $c_1$, ..., $c_k\in {\mathbb R}$, there exists an evaluation $e:\mathrm{Form}({\mathbb R}{\mathcal L})\rightarrow [0,1]$ such that

  $$\begin{aligned} \sum _{i=1}^k c_i(r_i-e(\varphi _i))\ge 0. \end{aligned}$$

Note that Lemma 12 is used to prove (3)$\Leftrightarrow$(4) and Lemma 9 (b) for (4)$\Leftrightarrow$(5). $\square$

If $r\in [0,1]$ and $\varphi$ is a formula then the piecewise linear function $r-\widetilde{\varphi }: [0,1]^n\rightarrow {\mathbb R}$ may have negative values, therefore it may not correspond to a formula of ${\mathbb R}{\mathcal L}$. The next result provides a necessary condition for a book to be coherent using quasi-linear combinations of formulas. We also prove a sufficient condition in Corollary 9, but using different formulas.

We set $\mathbf {r}=\Delta _r(\varphi \rightarrow \varphi )$ and $\varphi \ominus \psi =\varphi \odot \lnot \psi$ whenever $r\in [0,1]$ and $\varphi$, $\psi \in Form({\mathbb R}{\mathcal L})$. Note that $\chi =\varphi \ominus \psi$ implies $\widetilde{\chi }=0\vee (\widetilde{\varphi }-\widetilde{\psi })$.

Proposition 7

Assume $\varphi _1$, $\ldots$, $\varphi _k$ are formulas of ${\mathbb R}{\mathcal L}$ and $r_1$, $\ldots$, $r_k\in [0,1]$ such that the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent. Hence

$$\begin{aligned} qspan((\mathbf {r_1}\ominus \varphi _1),\ldots , (\mathbf {r_k}\ominus \varphi _k)) \end{aligned}$$

is a set of invalid formulas.

Proof

Let $\chi \in qspan((\mathbf {r_1}\ominus \varphi _1),\ldots , (\mathbf {r_k}\ominus \varphi _k) )$ and $c_1$, $\ldots$, $c_k\in {\mathbb R}$ such that

$$\begin{aligned} \widetilde{\chi }=\varrho \circ \left( \sum _{i=1}^kc_i(0\vee (r_i-\widetilde{\varphi _i}))\right) . \end{aligned}$$

Since the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent there is an evaluation $e$ such that

$$\begin{aligned}&\sum _{i=1}^k(-c_i)(r_i-e(\varphi _i))\ge 0, \,\,\mathrm{which\,\, implies\,\, that}\\&\sum _{i=1}^k(-c_i)((r_i-e(\varphi _i))\vee 0)\ge 0. \end{aligned}$$

We get $\sum _{i=1}^k(-c_i) e(\mathbf {r}_i \ominus \varphi _i)\ge 0$. By Lemma 9 (b), $\varrho (\sum _{i=1}^k c_i e(\mathbf {r_i}\ominus \varphi _i))=0$. Using Lemma 12 it follows that $e(\chi )=0$, so $\chi$ is an invalid formula. $\square$

If $\varphi \in Form({\mathbb R}{\mathcal L})$, $r\in [0,1]$ and $c\in {\mathbb R}$ we denote

$$\begin{aligned} \psi (\varphi ,r,c)=\left\{ \begin{array}{ll} \varphi \ominus \mathbf {r}, &{} \text{ if } c\ge 0\\ \mathbf {r}\ominus \varphi , &{} \text{ if } c < 0. \end{array}\right. \end{aligned}$$

Proposition 8

Assume $\varphi _1,\ldots ,\varphi _k\in Form({\mathbb R}{\mathcal L})$ and $r_1$, $\ldots$, $r_k\in [0,1]$ such that for any $c_1,\ldots ,c_k\in {\mathbb R}$ the formula $\Phi$ of ${\mathbb R}\mathcal{L}$ is invalid whenever $\widetilde{\Phi }= \varrho \circ (\sum _{i=1}^k|c_i|\widetilde{\psi _i})$, where $|c_i|$ is the module of $c_i$ and $\psi _i=\psi (\varphi _i,r_i,c_i)$ for any $i\in \{1,\ldots ,k\}$. Then the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent.

Proof

If $c_1$, $\cdots$, $c_k\in {\mathbb R}$ and $\Phi$ in $Form({\mathbb R}\mathcal{L})$ is an invalid formula such that $\widetilde{\Phi }= \varrho \circ (\sum _{i=1}^k|c_i|\widetilde{\psi _i})$, then there exists an evaluation $e$ such that $e(\Phi )=0$ and, by Lemma 12, we get $\varrho (\sum _{i=1}^k |c_i| e(\psi _i))=0$, so

$$\begin{aligned}&\varrho \left( \sum _{c_i\ge 0} c_i((e(\varphi _i)-r_i)\vee 0\right) \\&\quad +\sum _{c_i< 0} (-c_i)((r_i-e(\varphi _i))\vee 0) \left. \right) =0 \end{aligned}$$

By Lemma 9(a),

$$\begin{aligned} \varrho \left( \left( \sum _{c_i\ge 0} c_i(e(\varphi _i)-r_i) +\sum _{c_i< 0} (-c_i)(r_i-e(\varphi _i))\right) \vee 0\right) =0 \end{aligned}$$

and by Lemma 9(c),

$$\begin{aligned} \varrho \left( \sum _{c_i\ge 0} c_i(e(\varphi _i)-r_i) +\sum _{c_i< 0} (-c_i)(r_i-e(\varphi _i))\right) =0. \end{aligned}$$

Using Lemma 9 (b), we get

$$\begin{aligned} -\left( \sum _{c_i\ge 0} c_i(e(\varphi _i)-r_i) +\sum _{c_i< 0} (-c_i)(r_i-e(\varphi _i))\right) \ge 0. \end{aligned}$$

It follows that $\sum _{i=1}^k c_i (r_i-e(\varphi _i))\ge 0$. $\square$

Corollary 9

Assume $\varphi _1$, $\ldots$, $\varphi _k\in Form({\mathbb R}{\mathcal L})$ and $r_1$, $\ldots$, $r_k\in [0,1]$ such that

$$\begin{aligned} qspan(\alpha _1,\ldots ,\alpha _k,\beta _1,\ldots ,\beta _k) \end{aligned}$$

is a set of invalid formulas, where

$$\begin{aligned} \alpha _i=\mathbf {r}_i \ominus \varphi _i\,\, and\,\, \beta _i=\varphi _i\ominus \mathbf {r}_i \,\, for\,\, any\,\, i\,\, \in \{1,\ldots ,k\}. \end{aligned}$$

Then the book $\{(\varphi _i,r_i)\mid i\in \{1,\ldots ,k\}\}$ is coherent.

Proof

For any $c_1$, $\ldots$, $c_k\in {\mathbb R}$, if $\Phi$ is a formula of ${\mathbb R}\mathcal{L}$ such that $\widetilde{\Phi }= \varrho \circ \sum _{i=1}^k|c_i|\psi _i$ as in Proposition 8, then $\Phi \in qspan(\alpha _1,\ldots ,\alpha _k,\beta _1,\ldots ,\beta _k)$. In consequence, we can apply Proposition 8. $\square$

Remark 16

We initiate the theory of quasi-linear combinations in ${\mathbb R}{\mathcal L}$ and relate it with de Finetti’s notion of coherence, which can be expressed by an invalidity condition. The linear combinations of formulas from Łukasiewicz logic were approached in Castro and Trillas (1998) and Amato et al. (2005), as representations for a particular class of neural networks. The composition between the function $\varrho$ and a linear combination of formulas Łukasiewicz logic can be naturally represented by a formula in our logic ${\mathbb R}{\mathcal L},$ and therefore, the theory of linear combinations can be approached within a simple defined logical system. Note that ${\mathbb R}{\mathcal L}$ is a conservative extension of Łukasiewicz logic. It has standard completeness theorem with respect to $[0,1]$ and is supported by the algebraic theory of Riesz MV-algebras which are categorically equivalent to unital Riesz spaces. Hence, our hope for the future is that the system ${\mathbb R}{\mathcal L}$ is enough expressive for representing classes of neural networks in a pure logical frame, having in mind the role of classical logic in the synthesis and analysis of boolean circuits.