The Join of the Variety of MV-Algebras and the Variety of Orthomodular Lattices

Abstract

We axiomatize the smallest variety that contains both the variety of MV-algebras and the variety (term equivalent to the variety) of orthomodular lattices.

Effect algebras, which were introduced by Foulis and Bennett [8], are partially ordered partial algebras closely related to the logical foundations of quantum mechanics. The standard example is the structure of self-adjoint operators between zero and identity on a Hilbert space, the so-called effects. Also some commonly known (total) algebras, such as orthomodular lattices and MV-algebras, may be regarded as particular cases of effect algebras. We were not the first who observed that lattice-ordered effect algebras can in a natural way be made into total algebras 〈A;⊕, ^′,0,1〉 of type 〈2,1,0,0〉; see [2, 3]. We proved that the class $\mathcal {E}$ of such algebras is a variety containing both the variety $\mathcal {M}\mathcal {V}$ of MV-algebras and the variety $\mathcal {OM}$ (term equivalent to the variety) of orthomodular lattices; relative to $\mathcal {E}$, $\mathcal {M}\mathcal {V}$ and $\mathcal {OM}$ can be axiomatized by the identities x⊕y=y⊕x and x⊕x=x, respectively. We also observed that $\mathcal {E}$ is not the join of $\mathcal {M}\mathcal {V}$ and $\mathcal {OM}$ in the lattice of subvarieties of $\mathcal {E}$. The aim of the present paper is to axiomatize the join of the varieties $\mathcal {M}\mathcal {V}$ and $\mathcal {OM}$, but we actually axiomatize the join of $\mathcal {M}\mathcal {V}$ with any finitely based subvariety of $\mathcal {E}$.

First, we recall some basic facts about lattice effect algebras; for more information we refer the reader to the book [7]. An effect algebra is a structure 〈E;+,0,1〉, where + is a partial binary operation and 0,1 are two constants, satisfying the following conditions:

• (i) x+y=y+x if one side is defined;

• (ii) (x+y)+z=x+(y+z) if one side is defined;

• (iii) for every x∈E there exists a unique x ^′∈E such that x ^′+x=1;

• (iv) if x+1 is defined, then x=0.

This is the definition by Foulis and Bennett [8], but we should mention that effect algebras are essentially the same as weak orthoalgebras introduced by Giuntini and Greuling [9] who, however, attribute their introduction to Foulis and Randall. Besides, effect algebras are equivalent to D-posets which were introduced by Kôpka and Chovanec [14].

Every effect algebra is naturally ordered by stipulating that x≤y iff y=x+z for some z∈E; such an element z is unique and we may denote it by y−x. Thus, if x≤y, then y−x is the only element such that y=x+(y−x), or more directly, y−x=(x+y ^′)^′. The structure 〈E;≤,−,0,1〉 so obtained is a D-poset in the sense of [14].

It is worth noticing that x+y is defined iff x≤y ^′, in which case x+y=(y ^′−x)^′.

An effect algebra (or D-poset) which is a lattice with respect to the natural order ≤ is called a lattice effect algebra (or D-lattice).

Two examples of lattice effect algebras that are of particular interest to us are orthomodular lattices and MV-algebras. We refer the reader to [13] for orthomodular lattices, and to [6] for MV-algebras.

Orthomodular lattices are equivalent to lattice effect algebras satisfying the condition that x+x is defined only if x=0. Indeed, if 〈L;∨,∧, ^′,0,1〉 is an orthomodular lattice, then the structure 〈L;+,0,1〉, where x+y is defined and equals x∨y iff x≤y ^′, is a lattice effect algebra satisfying the condition; and conversely, if 〈E;+,0,1〉 is such a lattice effect algebra with induced lattice operations ∨ and ∧, then x+y=x∨y if x+y is defined, and 〈E;∨,∧, ^′,0,1〉 is an orthomodular lattice.

MV-algebras are equivalent to MV-effect algebras, i.e., lattice effect algebras satisfying (x∨y)−y=x−(x∧y) for all x,y. Though we do not want to go into details here, we need to mention that every MV-algebra 〈A;⊕, ^′,0,1〉 bears a lattice order which is given by x≤y iff x ^′⊕y=1 iff y=x⊕z for some z∈A. Now, if 〈A;⊕, ^′,0,1〉 is an MV-algebra, then 〈A;+,0,1〉, where x+y is defined and equals x⊕y iff x≤y ^′, is an MV-effect algebra; and conversely, if 〈E;+,0,1〉 is an MV-effect algebra, then by setting

$$ x\oplus y=(x\wedge y^{\prime})+y $$

(1)

we obtain an MV-algebra 〈E;⊕, ^′,0,1〉 with the same lattice order as 〈E;+,0,1〉.

Using (1), an arbitrary lattice effect algebra can be made into a total algebra. So let $\mathcal {E}$ be the class of all (total) algebras A=〈A;⊕, ^′,0,1〉 of type 〈2,1,0,0〉 that arise from lattice effect algebras by means of (1). It is useful to define two more total operations, as follows:

$$ x\oslash y=(x^{\prime}\oplus y)^{\prime} \quad\text{and}\quad x\ominus y=(y\oplus x^{\prime})^{\prime}. $$

(2)

In the language of lattice effect algebras we have

$$ x\oslash y=(x\vee y)-y \quad\text{and}\quad x\ominus y=x-(x\wedge y), $$

(3)

and it is straightforward to show that all algebras in $\mathcal {E}$ satisfy the identities

$$\begin{array}{@{}rcl@{}} &&x\oplus 0 = x, \end{array} $$

(4)

$$\begin{array}{@{}rcl@{}} &&x^{\prime\prime} = x, \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} &&(x\oslash y)\oplus y = (y\oslash x)\oplus x, \end{array} $$

(6)

$$\begin{array}{@{}rcl@{}} &&(((x\oplus y)\oslash y)\oplus z)^{\prime}\oplus (x\oplus z) = 1, \end{array} $$

(7)

and the quasi-identity

$$ x\oplus y\leq z^{\prime} \quad \Rightarrow\quad (x\oplus y)\oplus z = x\oplus (z\oplus y). $$

(8)

On the other hand, we proved in [3] that if an algebra A=〈A;⊕, ^′,0,1〉 satisfies the identities (4)–(7), then the rule x≤y iff x ^′⊕y=1 (equivalently, x⊘y=0 or x⊖y=0) defines a bounded lattice, with bounds 0 and 1, where

$$x\vee y=(x\oslash y)\oplus y \quad\text{and}\quad x\wedge y=(x^{\prime}\vee y^{\prime})^{\prime}=x\ominus (x\ominus y). $$

In [3], as well as in some other papers, we called these algebras “basic algebras”. Moreover, if A satisfies also (8), then the partial algebra 〈A;+,0,1〉 obtained by restricting ⊕ as in the case of MV-algebras, i.e., x+y is defined and equals x⊕y iff x≤y ^′, is a lattice effect algebra with the same natural order as A, which entails that the total algebra associated with 〈A;+,0,1〉 via (1) is the initial algebra A.

The quasi-identity (8) may be replaced with an identity; for example, it suffices to write (x⊕y)^′∧z instead of z. Hence the class $\mathcal {E}$ is a variety.

Recalling that MV-algebras correspond to MV-effect algebras, it is evident by (3) that the variety of MV-algebras $\mathcal {M}\mathcal {V}$ is the subvariety of $\mathcal {E}$ axiomatized by the identity x⊘y=x⊖y, which is in view of (2) equivalent to x⊕y=y⊕x. MV-algebras are usually defined as algebras A=〈A;⊕, ^′,0,1〉 such that 〈A;⊕,0〉 is a commutative monoid, satisfying the identities (5), (6) and 1⊕x=1=0^′; see [6].

We have seen that orthomodular lattices correspond to lattice effect algebras satisfying the condition that x≤x ^′ implies x=0. In the language of $\mathcal {E}$, this is equivalent to the identity x∧x ^′=0, and in turn to x⊕x=x. We let $\mathcal {OM}$ denote the subvariety of $\mathcal {E}$ defined by x∧x ^′=0 or x⊕x=x. Note that the total addition in orthomodular lattices is given by x⊕y=(x∧y ^′)∨y, which is not the same as x∨y.

The variety $\mathcal {E}$ is congruence distributive, because its members are lattice based algebras. In fact, $\mathcal {E}$ is an arithmetical variety; see [3]. Congruence distributivity implies that for any two subvarieties $\mathcal {V}_{1},\mathcal {V}_{2}$ of $\mathcal {E}$ one has $\text {Si}({\mathcal {V}_{1} \vee \mathcal {V}_{2}}) = \text {Si}({\mathcal {V}_{1}}) \cup \text {Si}({\mathcal {V}_{2}})$, where $\text {Si}(\mathcal {K})$ denotes the class of subdirectly irreducible algebras in the respective class $\mathcal {K}$. Since there exist subdirectly irreducible algebras in $\mathcal {E}$ which are neither in $\mathcal {M}\mathcal {V}$ nor in $\mathcal {OM}$, it follows that $\mathcal {E}$ is not the join $\mathcal {M}\mathcal {V}\vee \mathcal {OM}$. For instance, a non-trivial horizontal sum of non-Boolean MV-algebras is a simple algebra in $\mathcal {E}$, but it is neither an MV-algebra nor an orthomodular lattice.

The concept that plays a central role in our proof is compatibility (see [7], Sect. 1.10). In general, two elements x,y in an effect algebra are said to be compatible, in symbols x⇔y, if there exist x ₁,y ₁,z such that x=x ₁+z, y=y ₁+z and x ₁+z+y ₁ is defined. It is obvious that 0⇔x⇔1 and x⇔x ^′ for any x. Also, if x≤y or y≤x, then x⇔y. Another important fact that we will use repeatedly is that x⇔y iff x⇔y ^′. In lattice effect algebras we have

$$x\leftrightarrow y \quad\text{iff}\quad (x\vee y)-y=x-(x\wedge y) \quad\text{iff}\quad (x\vee y)-y\leq x \quad\text{iff}\quad x-(x\wedge y)\leq y^{\prime}, $$

and hence, in the language of the variety $\mathcal {E}$,

$$x\leftrightarrow y \quad\text{iff}\quad x\oslash y=x\ominus y \quad\text{iff}\quad x\oslash y\leq x \quad\text{iff}\quad x\ominus y\leq y^{\prime}. $$

Since x⇔y iff x⇔y ^′, it is also true that

$$x\leftrightarrow y \quad\text{iff}\quad x\oplus y=y\oplus x \quad\text{iff}\quad x\leq x\oplus y. $$

Consequently, since MV-effect algebras are lattice effect algebras where x⇔y for all x,y, the variety $\mathcal {M}\mathcal {V}$ may be axiomatized, relative to $\mathcal {E}$, by any of the identities x⊘y=x⊖y, x⊘y≤x, x⊖y≤y ^′, x⊕y=y⊕x or x≤x⊕y.

A block of an effect algebra is a maximal set of mutually compatible elements. For lattice effect algebras Riečanová [17] proved that if x,y are compatible with a given z, then so are x∨y, x∧y, x+y (when defined) and x−y (when defined). It follows that every block B of any algebra $\boldsymbol {A}\in \mathcal {E}$ is a subuniverse of A, and obviously, the subalgebra B is an MV-algebra. The intersection of the blocks of A is called the compatibility centre and we denote it by K(A). Clearly, K(A)={a∈A∣a⇔x for all x∈A} and K(A) is a subuniverse of A.

Though the addition ⊕ is neither commutative nor associative,¹ we may unambiguously write n⋅x=x⊕⋯⊕x (with n occurrences of x) for any positive integer $n\in \mathbb {N}$, because every element x belongs to a block which is in fact an MV-algebra. We will mostly write just nx instead of n⋅x.

By [3], the variety $\mathcal {E}$ is congruence regular, i.e., any congruence 𝜃 of an algebra $\boldsymbol {A}\in \mathcal {E}$ is determined by each of its classes [a] _𝜃 , and in particular, by [0] _𝜃 . We say that I⊆A is an ideal ² of A if I=[0] _𝜃 for some congruence 𝜃 of A. Pulmannová and Vinceková [16] proved that ∅≠I⊆A is an ideal if and only if

• (i) x⊕y∈I for all x,y∈I;

• (ii) x⊘y∈I for all x∈I and y∈A.

An alternative characterization of ideals can be found in [5]. Of course, if $\boldsymbol {A} \in \mathcal {M}\mathcal {V}$, then the condition (ii) amounts to saying that I is downwards closed.

For any ideal I of an algebra $\boldsymbol {A}\in \mathcal {E}$, the only congruence 𝜃(I) with the property that [0] _𝜃(I)=I is given by

$$\langle x,y\rangle\in\theta(I) \quad\text{iff}\quad x\oslash y,y\oslash x\in I \quad\text{iff}\quad x\ominus y,y\ominus x\in I. $$

When ordered by set inclusion, the ideals of $\boldsymbol {A}\in \mathcal {E}$ form a distributive lattice that is isomorphic to the congruence lattice of A under the mutually inverse assignments I↦𝜃(I) and 𝜃↦[0] _𝜃 .

In general, we do not have a reasonable description of the ideal Ig(X) generated by a given subset X⊆A, but if X is an ideal of the compatibility centre or X={a} where a∈K(A), then Ig(X) can be described easily:

FormalPara Lemma 1 (cf. [12], Prop. 1)

Let $\boldsymbol {A}\in \mathcal {E}$ . If J is an ideal of the MV-algebra K(A), then Ig(J)={x∈A∣x≤a for some a∈J}. In particular, for any a∈K(A), $\text {Ig}({a})=\{x\in A\mid x\leq na \text { for some}\,\, n\in \mathbb {N}\}$.

FormalPara Proof

The first part is but a translation of [12], Prop. 1, into the language of the variety $\mathcal {E}$. Since K(A) is an MV-algebra, it is obvious that for any a∈K(A), the ideal of K(A) generated by a is $\{x\in K(\boldsymbol {A})\mid x\leq na \text { for some}\,\, n\in \mathbb {N}\}$, and consequently, the ideal of A generated by a is $\{x\in A\mid x\leq na \text { for some}\,\, n\in \mathbb {N}\}$. □

We should notice that in any algebra $\boldsymbol {A}\in \mathcal {E}$ we have

$$ x\leq y \quad\Rightarrow\quad x\oplus z\leq y\oplus z,\quad x\oslash z\leq y\oslash z \quad\text{and}\quad z\ominus y\leq z\ominus x, $$

(9)

which follows directly from (1) and (3). In particular, we have y≤x⊕y, x⊘y≤y ^′ and x⊖y≤x. We will also need the identity

$$ (x\wedge y)\oplus z=(x\oplus z)\wedge (y\oplus z), $$

(10)

which is over $\mathcal {E}$ equivalent to the identities (x∨y)⊘z=(x⊘z)∨(y⊘z) and x⊖(y∧z)=(x⊖y)∨(x⊖z). It is not hard to show (cf. [7], Prop. 1.8.6) that any lattice effect algebra satisfies the equality (x∧y)+z=(x+z)∧(y+z) provided x+z,y+z are defined, whence it follows that the identity (10) holds in every algebra in $\mathcal {E}$. In fact, (9) and (10) hold in all “basic algebras” (i.e., algebras satisfying (4)–(7), see [15]).

FormalPara Lemma 2

Let $\boldsymbol {A}\in \mathcal {E}$ . If a ₁ ,…,a _n ,b∈A are such that a _i ⇔b and a _i ∧b=0 for each i=1,…,n, then s(a ₁ ,…,a _n )⇔b and s(a ₁ ,…,a _n )∧b=0 for every additive term ³ s(x ₁ ,…,x _n ).

FormalPara Proof

We know that if a _i ⇔b for every i=1,…,n, then also any “sum” of the a _i ’s is compatible with b. The rest is an easy induction: Suppose that s=p⊕q where p,q are additive terms for which the statement holds true. Let c=p(a ₁,…,a _n ) and d=q(a ₁,…,a _n ). Then b∧c=b∧d=0 and, since b⇔d and using (10), we have b∧s(a ₁,…,a _n )=b∧(c⊕d)=b∧(b⊕d)∧(c⊕d)=b∧((b∧c)⊕d)=b∧d=0. □

FormalPara Lemma 3

Let $\boldsymbol {A}\in \mathcal {E}$ and a∈A. Suppose that (x⊘(x⊕y))∧a=0 for all x,y∈A. Then a∈K(A) and the polar a ^⊥ ={x∈A∣x∧a=0} is an ideal of A such that Ig(a)∩a ^⊥ ={0}.

FormalPara Proof

We have x⊕y=y⊕x whenever x≤y ^′. Hence, by (8), x ^′⊕y≤a⊕(x ^′⊕y)=(a ^′⊘(x ^′⊕y))^′ implies

$$\begin{array}{@{}rcl@{}} ((x\oslash y)\wedge a)^{\prime} = (x^{\prime}\oplus y)\vee a^{\prime} = [a^{\prime}\oslash (x^{\prime}\oplus y)]\oplus (x^{\prime}\oplus y) \\ =(x^{\prime}\oplus y)\oplus [a^{\prime}\oslash (x^{\prime}\oplus y)] = x^{\prime}\oplus ([a^{\prime}\oslash (x^{\prime}\oplus y)]\oplus y), \end{array} $$

whence ((x⊘y)∧a)⊖x=x ^′⊘((x⊘y)∧a)^′=x ^′⊘(x ^′⊕z) where z=(a ^′⊘(x ^′⊕y))⊕y. But then ((x⊘y)∧a)⊖x≤a yields ((x⊘y)∧a)⊖x=(x ^′⊘(x ^′⊕z))∧a=0 by the assumption about a. Thus (x⊘y)∧a≤x for all x,y∈A.

Then x⊘a ^′=(x⊘a ^′)∧a≤x for any x∈A, because we have x⊘a ^′≤1⊘a ^′=a. This shows that a ^′∈K(A), whence also a∈K(A). Now, (i) if x,y∈a ^⊥, then x⊕y∈a ^⊥ by Lemma 2; and (ii) if x∈a ^⊥, then (x⊘y)∧a≤x∧a=0, so x⊘y∈a ^⊥. Hence a ^⊥ is an ideal of A.

Finally, if x∈Ig(a)∩a ^⊥, then x≤n a for some $n\in \mathbb {N}$ by Lemma 1, and x∧a=0, which entails x∧n a=0 by Lemma 2. Thus x=x∧n a=0. □

Now, let $\mathcal {K}$ be a finitely based subvariety of $\mathcal {E}$ incomparable with $\mathcal {M}\mathcal {V}$, and suppose that $\mathcal {K}$ is axiomatized, relative to $\mathcal {E}$, by an identity which is of the form

$$t(x_{1},\dots,x_{k}) = 0. $$

Since $\mathcal {K}$ is finitely based, it can always be axiomatized by a single identity of such a form. Indeed, any identity p=q can be replaced with p◇q=0 and q◇p=0 where ◇ is ⊖ or ⊘, and any finite family of identities t ₁=0,…,t _n =0 can be replaced with the identity t ₁∨⋯∨t _n =0.

FormalPara Theorem 4

If $\mathcal {K}$ is as above, then the join of $\mathcal {M}\mathcal {V}$ and $\mathcal {K}$ is axiomatized, relative to $\mathcal {E}$ , by the identity

$$ (x\oslash (x\oplus y))\wedge t(z_{1},\dots,z_{k}) = 0. $$

(11)

FormalPara Proof

Let $\mathcal {K}^{\prime }$ be the subvariety of $\mathcal {E}$ axiomatized by (11). Since x⊘(x⊕y)=0 holds in $\mathcal {M}\mathcal {V}$ and t(z ₁,…,z _k )=0 in $\mathcal {K}$, it is obvious that (11) holds in $\mathcal {M}\mathcal {V}$ as well as in $\mathcal {K}$. Hence $\mathcal {M}\mathcal {V}\vee \mathcal {K}\subseteq \mathcal {K}^{\prime }$.

Conversely, let $\boldsymbol {A}\in \text {Si}({\mathcal {K}^{\prime }})$ and suppose that $\boldsymbol {A}\notin \mathcal {K}$, i.e., there exist a ₁,…,a _k ∈A such that t(a ₁,…,a _k )≠0. Let b=t(a ₁,…,a _k ). Then (x⊘(x⊕y))∧b=0 for any x,y∈A, and so, by Lemma 3, b ^⊥ is an ideal such that Ig(b)∩b ^⊥={0}. Since A is a subdirectly irreducible algebra and Ig(b)≠{0}, we have b ^⊥={0}. But x⊘(x⊕y)∈b ^⊥ for all x,y∈A, hence A satisfies the identity x⊘(x⊕y)=0, i.e. x≤x⊕y. In other words, $\boldsymbol {A}\in \mathcal {M}\mathcal {V}$. Therefore, $\text {Si}({\mathcal {K}^{\prime }}) \subseteq \text {Si}({\mathcal {M}\mathcal {V}}) \cup \text {Si}({\mathcal {K}})=\text {Si}({\mathcal {M}\mathcal {V}\vee \mathcal {K}})$, whence $\mathcal {K}^{\prime }\subseteq \mathcal {M}\mathcal {V}\vee \mathcal {K}$. □

Let us return to orthomodular lattices. The variety $\mathcal {OM}$ is axiomatized, relative to $\mathcal {E}$, by the identity x∧x ^′=0, i.e., the term t(x ₁,…,x _k ) from the above theorem is t(x)=x∧x ^′, and hence by Theorem 4 we obtain:

FormalPara Corollary 5

The join of $\mathcal {M}\mathcal {V}$ and $\mathcal {OM}$ is axiomatized, relative to $\mathcal {E}$ , by the identity

$$(x\oslash (x\oplus y))\wedge z\wedge z^{\prime} = 0. $$

Let 〈D;+,0,1〉 be the so-called distributive diamond, i.e., the lattice effect algebra with universe D={0,a,b,1} such that a+a=1=b+b, while a+b and b+a are not defined. This effect algebra has a prominent role for distributive lattice effect algebras because by [10], every finite distributive lattice effect algebra is isomorphic to a direct product of chains and diamonds. Let D be the corresponding algebra in $\mathcal {E}$. Note that a⊕b=b and b⊕a=a. Relative to the variety $\mathcal {DE}$ of distributive members of $\mathcal {E}$, the variety V(D) generated by D was axiomatized by the identity (x⊖y)⊖2z=(x⊖2z)⊖(y⊖2z) in [11], and by 2x=3x in [4]. Using Theorem 4 we can simplify the axiomatization of the join of $\mathcal {M}\mathcal {V}$ and V(D) which was given in [1]. We may take the term t(x)=3x⊘2x.

FormalPara Corollary 6

The join of $\mathcal {M}\mathcal {V}$ and V( D) is axiomatized, relative to $\mathcal {DE}$ , by the identity

$$(x\oslash (x\oplus y))\wedge (3z\oslash 2z) = 0. $$