Distances in Algebra

Abstract

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property ($x \cdot e = e \cdot x = x$ for any x ∈ G), and the inverse property (for any x ∈ G, there exists an element x ⁻¹ ∈ G such that $x \cdot x^{-1} = x^{-1} \cdot x = e$).

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1 Group Metrics

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property ($x \cdot e = e \cdot x = x$ for any x ∈ G), and the inverse property (for any x ∈ G, there exists an element x ⁻¹ ∈ G such that $x \cdot x^{-1} = x^{-1} \cdot x = e$).

In additive notation, a group (G, +, 0) is a set G with a binary operation + such that the following properties hold: x + y ∈ G for any x, y ∈ G, $x + (y + z) = (x + y) + z$ for any x, y, z ∈ G, $x + 0 = 0 + x = x$ for any x ∈ G, and, for any x ∈ G, there exists an element − x ∈ G such that $x + (-x) = (-x) + x = 0$.

A group (G, ⋅ , e) is called finite if the set G is finite. A group (G, ⋅ , e) is called Abelian if it is commutative, i.e., x ⋅ y = y ⋅ x for any x, y ∈ G.

Most metrics considered in this section are group norm metrics on a group (G, ⋅ , e), defined by

$$\displaystyle{\vert \vert x \cdot y^{-1}\vert \vert }$$

(or, sometimes, by | | y ⁻¹ ⋅ x | | ), where | | . | | is a group norm, i.e., a function $\vert \vert.\vert \vert: G \rightarrow \mathbb{R}$ such that, for any x, y ∈ G, we have the following properties:

• 1.  | | x | | ≥ 0, with | | x | | = 0 if and only if x = e;

• 2. $\vert \vert x\vert \vert = \vert \vert x^{-1}\vert \vert$;

• 3.  | | x ⋅ y | | ≤ | | x | | + | | y | | (triangle inequality).

In additive notation, a group norm metric on a group (G, +, 0) is defined by $\vert \vert x + (-y)\vert \vert = \vert \vert x - y\vert \vert$, or, sometimes, by $\vert \vert (-y) + x\vert \vert$.

The simplest example of a group norm metric is the bi-invariant ultrametric (sometimes called the Hamming metric) | | x ⋅ y ⁻¹ | |  _H , where | | x | |  _H  = 1 for x ≠ e, and | | e | |  _H  = 0.

• Bi-invariant metric

  A metric (in general, a semimetric) d on a group (G, ⋅ , e) is called bi-invariant if

  $$\displaystyle{d(x,y) = d(x \cdot z,y \cdot z) = d(z \cdot x,z \cdot y)}$$

  for any x, y, z ∈ G (cf. translation invariant metric in Chap. 5). Any group norm metric on an Abelian group is bi-invariant.

  A metric (in general, a semimetric) d on a group (G, ⋅ , e) is called a right-invariant metric if d(x, y) = d(x ⋅ z, y ⋅ z) for any x, y, z ∈ G, i.e., the operation of right multiplication by an element z is a motion of the metric space (G, d). Any group norm metric defined by | | x ⋅ y ⁻¹ | | , is right-invariant.

  A metric (in general, a semimetric) d on a group (G, ⋅ , e) is called a left-invariant metric if d(x, y) = d(z ⋅ x, z ⋅ y) holds for any x, y, z ∈ G, i.e., the operation of left multiplication by an element z is a motion of the metric space (G, d). Any group norm metric defined by | | y ⁻¹ ⋅ x | | , is left-invariant.

  Any right-invariant or left-invariant (in particular, bi-invariant) metric d on G is a group norm metric, since one can define a group norm on G by | | x | | = d(x, 0).

• G-invariant metric

  Given a metric space (X, d) and an action g(x) of a group G on it, the metric d is called G-invariant (under this action) if for all x, y ∈ X, g ∈ G it holds

  $$\displaystyle{d(g(x),g(y)) = d(x,y).}$$

  For every G-invariant metric d _X on X and every point x ∈ X, the function

  $$\displaystyle{d_{G}(g_{1},g_{2}) = d_{X}(g_{1}(x),g_{2}(x))}$$

  is a left-invariant metric on G. This metric is called orbit metric in [BBI01], since it is the restriction of d on the orbit Gx, which can be identified with G.

• Positively homogeneous distance

  A distance d on an Abelian group (G, +, 0) is called positively homogeneous if

  $$\displaystyle{d(mx,my) = md(x,y)}$$

  for all x, y ∈ G and all $m \in \mathbb{N}$, where mx is the sum of m terms all equal to x.

• Translation discrete metric

  A group norm metric (in general, a group norm semimetric) on a group (G, ⋅ , e) is called translation discrete if the translation distances (or translation numbers)

  $$\displaystyle{\tau _{G}(x) =\lim _{n\rightarrow \infty }\frac{\vert \vert x^{n}\vert \vert } {n} }$$

  of the nontorsion elements x (i.e., such that x ⁿ ≠ e for any $n \in \mathbb{N}$) of the group with respect to that metric are bounded away from zero.

  If the numbers τ _G (x) are just nonzero, such a group norm metric is called a translation proper metric .

• Word metric

  Let (G, ⋅ , e) be a finitely-generated group with a set A of generators (i.e., A is finite, and every element of G can be expressed as a product of finitely many elements A and their inverses). The word length w  _W ^A(x) of an element x ∈ G∖{e} is defined by

  $$\displaystyle{w\,_{W}^{A}(x) =\inf \{ r: x = a_{ 1}^{\epsilon _{1} }\ldots a_{r}^{\epsilon _{r} },a_{i} \in A,\epsilon _{i} \in \{\pm 1\}\}\mbox{ and }w\,_{W}^{A}(e) = 0.}$$

  The word metric d _W ^A associated with A is a group norm metric on G defined by

  $$\displaystyle{w\,_{W}^{A}(x \cdot y^{-1}).}$$

  As the word length w  _W ^A is a group norm on G, d _W ^A is right-invariant. Sometimes it is defined as w  _W ^A(y ⁻¹ ⋅ x), and then it is left-invariant. In fact, d _W ^A is the maximal metric on G that is right-invariant, and such that the distance from any element of A or A ⁻¹ to the identity element e is equal to one.

  If A and B are two finite sets of generators of the group (G, ⋅ , e), then the identity mapping between the metric spaces (G, d _W ^A) and (G, d _W ^B) is a quasi-isometry, i.e., the word metric is unique up to quasi-isometry.

  The word metric is the path metric of the Cayley graph $\Gamma$ of (G, ⋅ , e), constructed with respect to A. Namely, $\Gamma$ is a graph with the vertex-set G in which two vertices x and y ∈ G are connected by an edge if and only if y = a ^ε x, ε = ±1, a ∈ A.

• Weighted word metric

  Let (G, ⋅ , e) be a finitely-generated group with a set A of generators. Given a bounded weight function w: A → (0, ∞), the weighted word length w  _WW ^A(x) of an element x ∈ G∖{e} is defined by w  _WW ^A(e) = 0 and

  $$\displaystyle{w\,_{WW}^{A}(x) =\inf \left \{\sum _{ i=1}^{t}w(a_{ i}),t \in \mathbb{N}: x = a_{1}^{\epsilon _{1} }\ldots a_{t}^{\epsilon _{t} },a_{i} \in A,\epsilon _{i} \in \{\pm 1\}\right \}.}$$

  The weighted word metric d _WW ^A associated with A is a group norm metric on G defined by

  $$\displaystyle{w\,_{WW}^{A}(x \cdot y^{-1}).}$$

  As the weighted word length w  _WW ^A is a group norm on G, d _WW ^A is right-invariant. Sometimes it is defined as w  _WW ^A(y ⁻¹ ⋅ x), and then it is left-invariant.

  The metric d _WW ^A is the supremum of semimetrics d on G with the property that d(e, a) ≤ w(a) for any a ∈ A.

  The metric d _WW ^A is a coarse-path metric, and every right-invariant coarse path metric is a weighted word metric up to coarse isometry.

  The metric d _WW ^A is the path metric of the weighted Cayley graph $\Gamma _{W}$ of (G, ⋅ , e) constructed with respect to A. Namely, $\Gamma _{W}$ is a weighted graph with the vertex-set G in which two vertices x and y ∈ G are connected by an edge with the weight w(a) if and only if y = a ^ε x, ε = ±1, a ∈ A.

• Interval norm metric

  An interval norm metric is a group norm metric on a finite group (G, ⋅ , e) defined by

  $$\displaystyle{\vert \vert x \cdot y^{-1}\vert \vert _{ int},}$$

  where | | . | |  _int is an interval norm on G, i.e., a group norm such that the values of | | . | |  _int form a set of consecutive integers starting with 0.

  To each interval norm | | . | |  _int corresponds an ordered partition $\{B_{0},\ldots,B_{m}\}$ of G with $B_{i} =\{ x \in G: \vert \vert x\vert \vert _{int} = i\}$; cf. Sharma–Kaushik distance in Chap. 16 The Hamming and Lee norms are special cases of interval norm. A generalized Lee norm is an interval norm for which each class has a form $B_{i} =\{ a,a^{-1}\}$.

• C-metric

  A C-metric d is a metric on a group (G, ⋅ , e) satisfying the following conditions:

  • 1. The values of d form a set of consecutive integers starting with 0;

  • 2. The cardinality of the sphere $B(x,r) =\{ y \in G: d(x,y) = r\}$ is independent of the particular choice of x ∈ G.

  The word metric, the Hamming metric, and the Lee metric are C-metrics. Any interval norm metric is a C-metric.

• Order norm metric

  Let (G, ⋅ , e) be a finite Abelian group. Let ord(x) be the order of an element x ∈ G, i.e., the smallest positive integer n such that x ⁿ = e. Then the function $\vert \vert.\vert \vert _{ord}: G \rightarrow \mathbb{R}$ defined by | | x | |  _ord  = lnord(x), is a group norm on G, called the order norm.

  The order norm metric is a group norm metric on G, defined by

  $$\displaystyle{\vert \vert x \cdot y^{-1}\vert \vert _{ ord}.}$$

• Tărnăuceanu metric

  Let o(a) denote the order of the element a of a group. Let C be the class of finite groups G in which o(ab) < o(a) + o(b) for every a, b ∈ G. Tărnăuceanu, 2015, noted that the function $d: G \times G \rightarrow \mathbb{N}$ defined by

  $$\displaystyle{d(x,y) = o(xy^{-1}) - 1}$$

  for all x, y ∈ G is a metric on G if and only if G ∈ C.

  He found that C contains all Abelian p-groups, Q ₈, and A ₄, but not nonabelian finite simple groups, alternating groups A(n) with n ≥ 5, and, for n ≥ 4, Sym(n), quaternion groups $Q_{2^{n}}$, dihedral groups D _2n . C is closed under subgroups, but not under direct products or extensions. The centralizers of nontrivial elements of such groups contain only elements of prime power order.

• Monomorphism norm metric

  Let (G, +, 0) be a group. Let (H, ⋅ , e) be a group with a group norm | | . | |  _H . Let f: G → H be a monomorphism of groups G and H, i.e., an injective function such that $f(x + y) = f(x) \cdot f(y)$ for any x, y ∈ G. Then the function $\vert \vert.\vert \vert _{G}^{f}: G \rightarrow \mathbb{R}$ defined by | | x | |  _G ^f = | | f(x) | |  _H , is a group norm on G, called the monomorphism norm.

  The monomorphism norm metric is a group norm metric on G defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{G}^{f}.}$$

• Product norm metric

  Let (G, +, 0) be a group with a group norm | | . | |  _G . Let (H, ⋅ , e) be a group with a group norm | | . | |  _H . Let $G \times H =\{\alpha = (x,y): x \in G,y \in H\}$ be the Cartesian product of G and H, and $(x,y) \cdot (z,t) = (x + z,y \cdot t)$.

  Then the function $\vert \vert.\vert \vert _{G\times H}: G \times H \rightarrow \mathbb{R}$ defined by $\vert \vert \alpha \vert \vert _{G\times H} = \vert \vert (x,y)\vert \vert _{G\times H} = \vert \vert x\vert \vert _{G} + \vert \vert y\vert \vert _{H}$, is a group norm on G × H, called the product norm.

  The product norm metric is a group norm metric on G × H defined by

  $$\displaystyle{\vert \vert \alpha \cdot \beta ^{-1}\vert \vert _{ G\times F}.}$$

  On the Cartesian product G × H of two finite groups with the interval norms | | . | |  _G ^int and | | . | |  _H ^int, an interval norm | | . | |  _G×H ^int can be defined. In fact, $\vert \vert \alpha \vert \vert _{G\times H}^{int} = \vert \vert (x,y)\vert \vert _{G\times H}^{int} = \vert \vert x\vert \vert _{G} + (m + 1)\vert \vert y\vert \vert _{H}$, where m = max _a ∈ G  | | a | |  _G ^int.

• Quotient norm metric

  Let (G, ⋅ , e) be a group with a group norm | | . | |  _G . Let (N, ⋅ , e) be a normal subgroup of (G, ⋅ , e), i.e., xN = Nx for any x ∈ G. Let (G∕N, ⋅ , eN) be the quotient group of G, i.e., $G/N =\{ xN: x \in G\}$ with xN = { x ⋅ a: a ∈ N}, and xN ⋅ yN = xyN. Then the function $\vert \vert.\vert \vert _{G/N}: G/N \rightarrow \mathbb{R}$ defined by $\vert \vert xN\vert \vert _{G/N} =\min _{a\in N}\vert \vert xa\vert \vert _{X}$, is a group norm on G∕N, called the quotient norm.

  A quotient norm metric is a group norm metric on G∕N defined by

  $$\displaystyle{\vert \vert xN \cdot (yN)^{-1}\vert \vert _{ G/N} = \vert \vert xy^{-1}N\vert \vert _{ G/N}.}$$

  If $G = \mathbb{Z}$ with the norm being the absolute value, and $N = m\mathbb{Z}$, $m \in \mathbb{N}$, then the quotient norm on $\mathbb{Z}/m\mathbb{Z} = \mathbb{Z}_{m}$ coincides with the Lee norm.

  If a metric d on a group (G, ⋅ , e) is right-invariant, then for any normal subgroup (N, ⋅ , e) of (G, ⋅ , e) the metric d induces a right-invariant metric (in fact, the Hausdorff metric ) d ^∗ on G∕N by

  $$\displaystyle{d^{{\ast}}(xN,yN) =\max \{\max _{ b\in yN}\min _{a\in xN}d(a,b),\max _{a\in xN}\min _{b\in yN}d(a,b)\}.}$$

• Commutation distance

  Let (G, ⋅ , e) be a finite nonabelian group. Let $Z(G) =\{ c \in G:\,\, x \cdot c = c \cdot x\,\,\mbox{ for any }\,\,x \in G\}$ be the center of G.

  The commutation graph of G is defined as a graph with the vertex-set G in which distinct elements x, y ∈ G are connected by an edge whenever they commute, i.e., x ⋅ y = y ⋅ x. (Darafsheh, 2009, consider noncommuting graph on G∖ Z(G).)

  Any two noncommuting elements x, y ∈ G are connected in this graph by the path x, c, y, where c is any element of Z(G) (for example, e). A path $x = x^{1},x^{2},\ldots,x^{k} = y$ in the commutation graph is called an (x − y) N-path if x ⁱ ∉ Z(G) for any $i \in \{ 1,\ldots,k\}$. In this case the elements x, y ∈ G∖Z(G) are called N-connected.

  The commutation distance (see [DeHu98]) d is an extended distance on G defined by the following conditions:

  • 1. d(x, x) = 0;

  • 2. d(x, y) = 1 if x ≠ y, and x ⋅ y = y ⋅ x;

  • 3. d(x, y) is the minimum length of an (x − y) N-path for any N-connected elements x and y ∈ G∖Z(G);

  • 4. d(x, y) = ∞ if x, y ∈ G∖Z(G) are not connected by any N-path.

  Given a group G and a G-conjugacy class X in it, Bates–Bundy–Perkins–Rowley in 2003, 2004, 2007, 2008 considered commuting graph (X, E) whose vertex set is X and distinct vertices x, y ∈ X are joined by an edge e ∈ E whenever they commute.

• Modular distance

  Let $(\mathbb{Z}_{m},+,0)$, m ≥ 2, be a finite cyclic group. Let $r \in \mathbb{N}$, r ≥ 2. The modular r-weight w _r (x) of an element $x \in \mathbb{Z}_{m} =\{ 0,1,\ldots,m\}$ is defined as $w_{r}(x) =\min \{ w_{r}(x),w_{r}(m - x)\}$, where w _r (x) is the arithmetic r-weight of the integer x.

  The value w _r (x) can be obtained as the number of nonzero coefficients in the generalized nonadjacent form $x = e_{n}r^{n} +\ldots e_{1}r + e_{0}$ with $e_{i} \in \mathbb{Z}$, | e _i  | < r, $\vert e_{i} + e_{i+1}\vert <r$, and | e _i  | < | e _i+1 | if e _i e _i+1 < 0. Cf. arithmetic r-norm metric in Chap. 12

  The modular distance is a distance on $\mathbb{Z}_{m}$, defined by

  $$\displaystyle{w_{r}(x - y).}$$

  The modular distance is a metric for w _r (m) = 1, w _r (m) = 2, and for several special cases with w _r (m) = 3 or 4. In particular, it is a metric for m = r ⁿ or $m = r^{n} - 1$; if r = 2, it is a metric also for $m = 2^{n} + 1$ (see, for example, [Ernv85]).

  The most popular metric on $\mathbb{Z}_{m}$ is the Lee metric defined by | | x − y | |  _Lee , where $\vert \vert x\vert \vert _{Lee} =\min \{ x,m - x\}$ is the Lee norm of an element $x \in \mathbb{Z}_{m}$.

• G-norm metric

  Consider a finite field $\mathbb{F}_{p^{n}}$ for a prime p and a natural number n. Given a compact convex centrally-symmetric body G in $\mathbb{R}^{n}$, define the G-norm of an element $x \in \mathbb{F}_{p^{n}}$ by $\vert \vert x\vert \vert _{G} =\inf \{\mu \geq 0: x \in p\mathbb{Z}^{n} +\mu G\}$.

  The G-norm metric is a group norm metric on $\mathbb{F}_{p^{n}}$ defined by

  $$\displaystyle{\vert \vert x \cdot y^{-1}\vert \vert _{ G}.}$$

• Permutation norm metric

  Given a finite metric space (X, d), the permutation norm metric is a group norm metric on the group (Sym _X , ⋅ , id) of all permutations of X (id is the identity mapping) defined by

  $$\displaystyle{\vert \vert \,f \cdot g^{-1}\vert \vert _{ Sym},}$$

  where the group norm | | . | |  _Sym on Sym _X is given by | | f | |  _Sym  = max _x ∈ X d(x, f(x)).

• Metric of motions

  Let (X, d) be a metric space, and let p ∈ X be a fixed element of X.

  The metric of motions (see [Buse55]) is a metric on the group $(\Omega,\cdot,id)$ of all motions of (X, d) (id is the identity mapping) defined by

  $$\displaystyle{\sup _{x\in X}d(\,f(x),g(x)) \cdot e^{-d(\,p,x)}}$$

  for any $f,g \in \Omega$ (cf. Busemann metric of sets in Chap. 3). If the space (X, d) is bounded, a similar metric on $\Omega$ can be defined as

  $$\displaystyle{\sup _{x\in X}d(\,f(x),g(x)).}$$

  Given a semimetric space (X, d), the semimetric of motions on $(\Omega,\cdot,id)$ is

  $$\displaystyle{d(\,f(\,p),g(\,p)).}$$

• General linear group semimetric

  Let $\mathbb{F}$ be a locally compact nondiscrete topological field. Let $(\mathbb{F}^{n},\vert \vert.\vert \vert _{\mathbb{F}^{n}})$, n ≥ 2, be a normed vector space over $\mathbb{F}$. Let | | . | | be the operator norm associated with the normed vector space $(\mathbb{F}^{n},\vert \vert.\vert \vert _{\mathbb{F}^{n}})$. Let $GL(n, \mathbb{F})$ be the general linear group over $\mathbb{F}$. Then the function $\vert.\vert _{op}: GL(n, \mathbb{F}) \rightarrow \mathbb{R}$ defined by $\vert g\vert _{op} =\sup \{ \vert \ln \vert \vert g\vert \vert \,\vert,\vert \ln \vert \vert g^{-1}\vert \vert \,\vert \}$, is a seminorm on $GL(n, \mathbb{F})$.

  The general linear group semimetric on the group $GL(n, \mathbb{F})$ is defined by

  $$\displaystyle{\vert g \cdot h^{-1}\vert _{ op}.}$$

  It is a right-invariant semimetric which is unique, up to coarse isometry, since any two norms on $\mathbb{F}^{n}$ are bi-Lipschitz equivalent.

• Generalized torus semimetric

  Let (T, ⋅ , e) be a generalized torus, i.e., a topological group which is isomorphic to a direct product of n multiplicative groups $\mathbb{F}_{i}^{{\ast}}$ of locally compact nondiscrete topological fields $\mathbb{F}_{i}$. Then there is a proper continuous homomorphism $v: T \rightarrow \mathbb{R}^{n}$, namely, $v(x_{1},\ldots,x_{n}) = (v_{1}(x_{1}),\ldots,v_{n}(x_{n}))$, where $v_{i}: \mathbb{F}_{i}^{{\ast}}\rightarrow \mathbb{R}$ are proper continuous homomorphisms from the $\mathbb{F}_{i}^{{\ast}}$ to the additive group $\mathbb{R}$, given by the logarithm of the valuation. Every other proper continuous homomorphism $v^{'}: T \rightarrow \mathbb{R}^{n}$ is of the form v ^′ = α ⋅ v with $\alpha \in GL(n, \mathbb{R})$. If | | . | | is a norm on $\mathbb{R}^{n}$, one obtains the corresponding seminorm | | x | |  _T  = | | v(x) | | on T.

  The generalized torus semimetric is defined on the group (T, ⋅ , e) by

  $$\displaystyle{\vert \vert xy^{-1}\vert \vert _{ T} = \vert \vert v(xy^{-1})\vert \vert = \vert \vert v(x) - v(y)\vert \vert.}$$

• Stable norm metric

  Given a Riemannian manifold (M, g), the stable norm metric is a group norm metric on its real homology group $H_{k}(M, \mathbb{R})$ defined by the following stable norm | | h | |  _s : the infimum of the Riemannian k-volumes of real cycles representing h.

  The Riemannian manifold $(\mathbb{R}^{n},g)$ is within finite Gromov–Hausdorff distance (cf. Chap. 1) from an n-dimensional normed vector space $(\mathbb{R}^{n},\vert \vert.\vert \vert _{s})$.

  If (M, g) is a compact connected oriented Riemannian manifold, then the manifold $H_{1}(M, \mathbb{R})/H_{1}(M, \mathbb{R})$ with metric induced by | | . | |  _s is called the Albanese torus (or Jacobi torus) of (M, g). This Albanese metric is a flat metric (Chap. 8).

• Heisenberg metric

  Let (H, ⋅ , e) be the (real) Heisenberg group $\mathcal{H}^{n}$, i.e., a group on the set $H = \mathbb{R}^{n} \times \mathbb{R}^{n} \times \mathbb{R}$ with the group law $h \cdot h' = (x,y,t) \cdot (x',y',t') = (x + x',y + y',t + t' + 2\sum _{i=1}^{n}(x_{i}'y_{i} - x_{i}y_{i}')$, and the identity e = (0, 0, 0). Let | . |  _Heis be the Heisenberg gauge (Cygan, 1978) on $\mathcal{H}^{n}$ defined by | h |  _Heis  = | (x, y, t) |  _Heis  = ((∑ _i = 1 ⁿ(x _i ² + y _i ²))² + t ²)^1∕4.

  The Heisenberg metric (or Korányi metric , Cygan metric , gauge metric ) d _Heis is a group norm metric on $\mathcal{H}^{n}$ defined by

  $$\displaystyle{\vert x^{-1} \cdot y\vert _{ Heis}.}$$

  One can identify the Heisenberg group $\mathcal{H}^{n-1} = \mathbb{C}^{n-1} \times \mathbb{R}$ with $\partial \mathbb{H}_{\mathbb{C}}^{n}\setminus \{\infty \}$, where $\mathbb{H}_{\mathbb{C}}^{n}$ is the Hermitian (i.e., complex) hyperbolic n-space, and ∞ is any point of its boundary $\partial \mathbb{H}_{\mathbb{C}}^{n}$. So, the usual hyperbolic metric of $\mathbb{H}_{\mathbb{C}}^{n+1}$ induces a metric on $\mathcal{H}^{n}$. The Hamenstädt distance on $\partial \mathbb{H}_{\mathbb{C}}^{n}\setminus \{\infty \}$ (Hersonsky–Paulin, 2004) is $\frac{1} {\sqrt{2}}d_{Heis}$.

  Sometimes, the term Cygan metric is reserved for the extension of the metric d _Heis on whole $\mathbb{H}_{\mathbb{C}}^{n}$ and (Apanasov, 2004) for its generalization (via the Carnot group $\mathbb{F}^{n-1} \times Im\mathbb{F}$) on $\mathbb{F}$-hyperbolic spaces $\mathbb{H}_{\mathbb{F}}^{n}$ over numbers $\mathbb{F}$ that can be complex numbers, or quaternions or, for n = 2, octonions. Also, the generalization of d _Heis on Carnot groups of Heisenberg type is called the Cygan metric.

  The second natural metric on $\mathcal{H}^{n}$ is the Carnot–Carathéodory metric (or CC metric, sub-Riemannian metric; cf. Chap. 7) d _C defined as the length metric (Chap. 6) using horizontal vector fields on $\mathcal{H}^{n}$. This metric is the internal metric (Chap. 4) corresponding to d _Heis .

  The metric d _Heis is bi-Lipschitz equivalent with d _C but not with any Riemannian distance and, in particular, not with any Euclidean metric. For both metrics, the Heisenberg group $\mathcal{H}^{n}$ is a fractal since its Hausdorff dimension, 2n + 2, is strictly greater than its topological dimension, 2n + 1.

• Metric between intervals

  Let G be the set of all intervals [a, b] of $\mathbb{R}$. The set G forms semigroups (G, +) and (G, ⋅ ) under addition $I + J =\{ x + y: x \in I,y \in J\}$ and under multiplication I ⋅ J = { x ⋅ y: x ∈ I, y ∈ J}, respectively.

  The metric between intervals is a metric on G, defined by

  $$\displaystyle{\max \{\vert I\vert,\vert J\vert \}}$$

  for all I, J ∈ G, where, for K = [a, b], one has $\vert K\vert = \vert a - b\vert$.

• Metric between games

  Consider positional games, i.e., two-player nonrandom games of perfect information with real-valued outcomes. Play is alternating with a nonterminated game having move options for both players. Real-world examples include Chess, Go and Tic-Tac-Toe. Formally, let $F_{\mathbb{R}}$ be the universe of games defined inductively as follows:

  • 1. Every real number $r \in \mathbb{R}$ belongs to $F_{\mathbb{R}}$ and is called an atomic game.

  • 2. If $A,B \subset F_{\mathbb{R}}$ with 1 ≤ | A | , | B | < ∞, then $\{A\vert B\} \in F_{\mathbb{R}}$ (nonatomic game).

  Write any game G = { A | B} as {G ^L | G ^R}, where G ^L = A and G ^R = B are the set of left and right moves of G, respectively.

  $F_{\mathbb{R}}$ becomes a commutative semigroup under the following addition operation:

  • 1. If p and q are atomic games, then p + q is the usual addition in $\mathbb{R}$.

  • 2. $p +\{ g_{l_{1}},\ldots \vert g_{r_{1}},\ldots \}=\{ g_{l_{1}} + p,\ldots \vert g_{r_{1}} + p,\ldots \}$.

  • 3. If G and H are both nonatomic, then $\{G^{L}\vert G^{R}\} +\{ H^{L}\vert H^{R}\} =\{ I^{L}\vert I^{R}\}$, where $I^{L} =\{ g_{l} + H,G + h_{l}: g_{l} \in G^{L},h_{l} \in H^{L}\}$ and $I^{R} =\{ g_{r} + H,G + h_{r}: g_{r} \in G^{R},h_{r} \in H^{R}\}$.

  For any game $G \in F_{\mathbb{R}}$, define the optimal outcomes $\overline{L}(G)$ and $\overline{R}(G)$ (if both players play optimally with Left and Right starting, respectively) as follows:

  $\overline{L}(\,p) = \overline{R}(\,p) = p$ and $\overline{L}(G) =\max \{ \overline{R}(g_{l}): g_{l} \in G^{L}\}$, $\overline{R}(G) =\max \{ \overline{L}(g_{r}): g_{r} \in G^{R}\}$.

  The metric between games G and H defined by Ettinger, 2000, is the following extended metric on $F_{\mathbb{R}}$:

  $$\displaystyle{\sup _{X}\vert \overline{L}(G + X) -\overline{L}(H + X)\vert =\sup _{X}\vert \overline{R}(G + X) -\overline{R}(H + X)\vert.}$$

• Helly semimetric

  Consider a game $(\mathcal{A},\mathcal{B},H)$ between players A and B with strategy sets $\mathcal{A}$ and $\mathcal{B}$, respectively. Here H = H(⋅ , ⋅ ) is the payoff function, i.e., if player A plays $a \in \mathcal{A}$ and player B plays $b \in \mathcal{B}$, then A pays H(a,b) to B. A player’s strategy set is the set of available to him pure strategies, i.e., complete algorithms for playing the game, indicating the move for every possible situation throughout it.

  The Helly semimetric between strategies $a_{1} \in \mathcal{A}$ and $a_{2} \in \mathcal{A}$ of A is defined by

  $$\displaystyle{\sup _{b\in \mathcal{B}}\vert H(a_{1},b) - H(a_{2},b)\vert.}$$

• Factorial ring semimetric

  Let (A, +, ⋅ ) be a factorial ring, i.e., an integral domain (nonzero commutative ring with no nonzero zero divisors), in which every nonzero nonunit element can be written as a product of (nonunit) irreducible elements, and such factorization is unique up to permutation.

  The factorial ring semimetric is a semimetric on the set A∖{0}, defined by

  $$\displaystyle{\ln \frac{lcm(x,y)} {gcd(x,y)},}$$

  where lcm(x, y) is the least common multiple, and gcd(x, y) is the greatest common divisor of elements x, y ∈ A∖{0}.

• Frankild–Sather–Wagstaff metric

  Let $\mathcal{G}(R)$ be the set of isomorphism classes, up to a shift, of semidualizing complexes over a local Noetherian commutative ring R. An R-complex is a particular sequence of R-module homomorphisms; see [FrSa07]) for exact definitions.

  The Frankild–Sather–Wagstaff metric ([FrSa07]) is a metric on $\mathcal{G}(R)$ defined, for any classes $[K],[L] \in \mathcal{G}(R)$, as the infimum of the lengths of chains of pairwise comparable elements starting with [K] and ending with [L].

2 Metrics on Binary Relations

A binary relation R on a set X is a subset of X × X; it is the arc-set of the directed graph (X, R) with the vertex-set X.

A binary relation R which is symmetric ((x, y) ∈ R implies (y, x) ∈ R), reflexive (all (x, x) ∈ R), and transitive ((x, y), (y, z) ∈ R imply (x, z) ∈ R) is called an equivalence relation or a partition (of X into equivalence classes). Any q-ary sequence x = (x ₁, …, x _n ), q ≥ 2 (i.e., with 0 ≤ x _i  ≤ q − 1 for 1 ≤ i ≤ n), corresponds to the partition $\{B_{0},\ldots,B_{q-1}\}$ of $V _{n} =\{ 1,\ldots,n\}$, where $B_{j} =\{ 1 \leq i \leq n: x_{i} = j\}$ are the equivalence classes.

A binary relation R which is antisymmetric ((x, y), (y, x) ∈ R imply x = y), reflexive, and transitive is called a partial order, and the pair (X, R) is called a poset (partially ordered set). A partial order R on X is denoted also by ⪯ with x ⪯ y if and only if (x, y) ∈ R. The order ⪯ is called linear if any elements x, y ∈ X are compatible, i.e., x ⪯ y or y ⪯ x.

A poset (L, ⪯) is called a lattice if every two elements x, y ∈ L have the join x ∨ y and the meet x ∧ y. All partitions of X form a lattice $\mathbb{P}_{X}$ by refinement; it is a sublattice of the lattice (by set-inclusion) of all binary relations.

• Kemeny distance

  The Kemeny distance between binary relations R ₁ and R ₂ on a set X is the Hamming metric | R ₁△R ₂ | . It is twice the minimal number of inversions of pairs of adjacent elements of X which is necessary to obtain R ₂ from R ₁.

  If R ₁, R ₂ are partitions, then the Kemeny distance coincides with the Mirkin–Tcherny distance , and $1 -\frac{\vert R_{1}\bigtriangleup R_{2}\vert } {n(n-1)}$ is the Rand index.

  If binary relations R ₁, R ₂ are linear orders (or permutations) on the set X, then the Kemeny distance coincides with the Kendall τ distance (Chap. 11).

• Drápal–Kepka distance

  The Drápal–Kepka distance between distinct quasigroups (differing from groups in that they need not be associative) (X, +) and (X, ⋅ ) is the Hamming metric | {(x, y): x + y ≠ x ⋅ y} | between their Cayley tables.

  For finite nonisomorphic groups, this distance is (Ivanyos, Le Gall and Yoshida, 2012) at least $2(\frac{\vert X\vert } {3} )^{2}$ with equality (Drápal, 2003) for some 3-groups.

• Editing metrics between partitions

  Let X be a finite set, | X | = n, and let A, B be nonempty subsets of X. Let $\mathcal{P}_{X}$ be the set of partitions of X, and $P,Q \in \mathcal{P}_{X}$. Let $P_{1},\ldots,P_{q}$ be blocks in the partition P, i.e., the pairwise disjoint sets such that $X = P_{1} \cup \ldots \cup P_{q}$, q ≥ 1. Let P ∨ Q and P ∧ Q be the join and meet of P and Q in the lattice $\mathbb{P}_{X}$ of partitions of X.

  Consider the following editing operations on partitions (clusterings):

  • An augmentation transforms a partition P of A∖{B} into a partition of A by either including the objects of B in a block, or including B as a new block;

  • An removal transforms a partition P of A into a partition of A∖{B} by deleting the objects in B from each block that contains them;

  • A division transforms one partition P into another by the simultaneous removal of B from P _i (where B ⊂ P _i , B ≠ P _i ), and augmentation of B as a new block;

  • A merging transforms one partition P into another by the simultaneous removal of B from P _i (where B = P _i ), and augmentation of B to P _j (where j ≠ i);

  • A transfer transforms one partition P into another by the simultaneous removal of B from P _i (where B ⊂ P _i ), and augmentation of B to P _j (where j ≠ i).

  Define (see, say, [Day81]), using above operations, the following metrics on $\mathcal{P}_{X}$:

  • 1. The minimum number of augmentations and removals of single objects needed to transform P into Q;

  • 2. The minimum number of divisions, mergings, and transfers of single objects needed to transform P into Q;

  • 3. The minimum number of divisions, mergings, and transfers needed to transform P into Q;

  • 4. The minimum number of divisions and mergings needed to transform P into Q; in fact, it is equal to $\vert P\vert + \vert Q\vert - 2\vert P \vee Q\vert$;

  • 5. $\sigma (\,P) +\sigma (Q) - 2\sigma (\,P \wedge Q)$, where $\sigma (\,P) =\sum _{P_{i}\in P}\vert P_{i}\vert (\vert P_{i}\vert - 1)$;

  • 6. $e(\,P) + e(Q) - 2e(\,P \wedge Q)$, where $e(\,P) =\log _{2}n +\sum _{P_{i}\in P}\frac{\vert P_{i}\vert } {n} \log _{2}\frac{\vert P_{i}\vert } {n}$;

  • 7. $2n -\sum _{P_{i}\in P}\max _{Q_{j}\in Q}\vert P_{i} \cap Q_{j}\vert -\sum _{Q_{j}\in Q}\max _{P_{i}\in P}\vert P_{i} \cap Q_{j}\vert$ (van Dongen, 2000).

  The maximum matching distance (or partition-distance as defined in Gusfield, 2002) is (Réignier, 1965) the minimum number of elements that must be moved between the blocks of partition P in order to transform it into Q.

• Rossi–Hamming metric

  Given a partition $P = (\,P_{1},\ldots,P_{q})$ of a finite set X, its size is defined as $s(\,P) = \frac{1} {2}\sum _{1\leq i\leq q}\vert P_{i}\vert (\vert P_{i}\vert - 1)$. We call the Rossi–Hamming metric the metric between partitions P and Q, defined in Rossi, 2014, as

  $$\displaystyle{d_{RH}(\,P,Q) = s(\,P) + s(Q) - 2s(\,P \wedge Q).}$$

  One has d _RH ( P, Q) ≤ s( P ∨ Q) − s( P ∧ Q), where the right-hand side is the size-based distance (Rossi, 2011). The inequality is strict only for some noncomparable P, Q.

3 Metrics on Semilattices

Consider a poset (L, ⪯). The meet (or infimum) x ∧ y (if it exists) of two elements x and y is the unique element satisfying x ∧ y ⪯ x, y, and z ⪯ x ∧ y if z ⪯ x, y. The join (or supremum) x ∨ y (if it exists) is the unique element such that x, y ⪯ x ∨ y, and x ∨ y ⪯ z if x, y ⪯ z. A poset (L, ⪯) is called a lattice if every its elements x, y have the join x ∨ y and the meet x ∧ y. A poset is a meet (or lower) semilattice if only the meet-operation is defined. A poset is a join (or upper) semilattice if only the join-operation is defined.

A lattice $\mathbb{L} = (L,\preceq,\vee,\wedge )$ is called a semimodular lattice if the modularity relation x M y is symmetric: xMy implies yMx for any x, y ∈ L. Here two elements x and y are said to constitute a modular pair, in symbols xMy, if x ∧ (y ∨ z) = (x ∧ y) ∨ z for any z ⪯ x. A lattice $\mathbb{L}$ in which every pair of elements is modular, is called a modular lattice.

Given a lattice $\mathbb{L}$, a function $v: L \rightarrow \mathbb{R}_{\geq 0}$, satisfying $v(x \vee y) + v(x \wedge y) \leq v(x) + v(y)$ for all x, y ∈ L, is called a subvaluation on $\mathbb{L}$. A subvaluation v is isotone if v(x) ≤ v(y) whenever x⪯ y, and it is positive if v(x) < v(y) whenever x⪯ y, x ≠ y. A subvaluation v is called a valuation if it is isotone and $v(x \vee y) + v(x \wedge y) = v(x) + v(y)$ for all x, y ∈ L.

• Lattice valuation metric

  Let $\mathbb{L} = (L,\preceq,\vee,\wedge )$ be a lattice, and let v be an isotone subvaluation on $\mathbb{L}$. The lattice subvaluation semimetric d _v on L is defined by

  $$\displaystyle{2v(x \vee y) - v(x) - v(y).}$$

  (It can be defined also on some semilattices.) If v is a positive subvaluation on $\mathbb{L}$, one obtains a metric, called the lattice subvaluation metric. If v is a valuation, d _v is called the valuation semimetric and can be written as

  $$\displaystyle{v(x \vee y) - v(x \wedge y) = v(x) + v(y) - 2v(x \wedge y).}$$

  If v is a positive valuation on $\mathbb{L}$, one obtains a metric, called the lattice valuation metric, and the lattice is called a metric lattice .

  An example is the Hamming distance $d_{H}(A,B) = \vert A \cup B\vert -\vert A \cap B\vert$ on the lattice ( P(X), ∪, ∩) of all subsets of the set X. Cf. also the Shannon distance (Chap. 14), which can be seen as a distance on partitions.

  If $L = \mathbb{N}$ (the set of positive integers), x ∨ y = lcm(x, y) (least common multiple), x ∧ y = gcd(x, y) (greatest common divisor), and the positive valuation v(x) = lnx, then $d_{v}(x,y) =\ln \frac{lcm(x,y)} {gcd(x,y)}$.

  This metric can be generalized on any factorial ring equipped with a positive valuation v such that v(x) ≥ 0 with equality only for the multiplicative unit of the ring, and $v(xy) = v(x) + v(y)$. Cf. factorial ring semimetric.

• Finite subgroup metric

  Let (G, ⋅ , e) be a group. Let $\mathbb{L} = (L,\subset,\cap )$ be the meet semilattice of all finite subgroups of the group (G, ⋅ , e) with the meet X ∩ Y and the valuation v(X) = ln | X | .

  The finite subgroup metric is a valuation metric on L defined by

  $$\displaystyle{v(X) + v(Y ) - 2v(X \wedge Y ) =\ln \frac{\vert X\vert \vert Y \vert } {(\vert X \cap Y \vert )^{2}}.}$$

• Join semilattice distances

  Let $\mathbb{L} = (L,\preceq,\vee )$ be a join semilattice, finite or infinite, such that every maximal chain in every interval [x, y] is finite. For x⪯ y, the height h(x, y) of y above x is the least cardinality of a finite maximal (by inclusion) chain of [x, y] minus 1. Call the join semilattice $\mathbb{L}$ semimodular if for all x, y ∈ L, whenever there exists an element z covered by both x and y, the join x ∨ y covers both x and y, or, in other words, whenever elements x, y have a common lower bound z, it holds h(x, x ∨ y) ≤ h(z, y). Any tree (i.e., all intervals [x, z] are finite, each pair x, y of uncomparable elements have a least common upper bound x ∨ y but they never have a common lower bound) is semimodular. Consider the following distances on L:

  d _path(x, y) is the path metric of the Hasse diagram of (L, ⪯), i.e., a graph with vertex-set L and an edge between two elements if they are comparable.

  d _a. path(x, y) is the smallest number of the form h(x, z) + h(y, z), where z is a common upper bound of x and y, i.e., it is the ancestral path distance; cf. pedigree-based distances in Chap. 23 This and next distance reflect the way how Roman civil law and medieval canon law, respectively, measured degree of kinship.

  d _max(x, y) is defined by max(h(x, x ∨ y), h(y, x ∨ y)).

  It holds d _a. path(x, y) ≥ d _path(x, y) ≥ d _max(x, y). Foldes, 2013, proved that d _max(x, y) is a metric if $\mathbb{L}$ is semimodular and that d _a. path(x, y) is a metric if and only if $\mathbb{L}$ is semimodular, in which case d _a. path(x, y) = d _path(x, y).

• Gallery distance of flags

  Let $\mathbb{L}$ be a lattice. A chain C in $\mathbb{L}$ is a subset of L which is linearly ordered, i.e., any two elements of C are compatible. A flag is a chain in $\mathbb{L}$ which is maximal with respect to inclusion. If $\mathbb{L}$ is a semimodular lattice, containing a finite flag, then $\mathbb{L}$ has a unique minimal and a unique maximal element, and any two flags C, D in $\mathbb{L}$ have the same cardinality, n + 1. Then n is the height of the lattice $\mathbb{L}$.

  Two flags C, D are called adjacent if either they are equal or D contains exactly one element not in C. A gallery from C to D of length m is a sequence of flags $C = C_{0},C_{1},\ldots,C_{m} = D$ such that C _i−1 and C _i are adjacent for $i = 1,\ldots,m$.

  A gallery distance of flags (see [Abel91]) is a distance on the set of all flags of a semimodular lattice $\mathbb{L}$ with finite height defined as the minimum of lengths of galleries from C to D. It can be written as

  $$\displaystyle{\vert C \vee D\vert -\vert C\vert = \vert C \vee D\vert -\vert D\vert,}$$

  where C ∨ D = { c ∨ d: c ∈ C, d ∈ D} is the subsemilattice generated by C and D. This distance is the gallery metric of the chamber system consisting of flags.

• Scalar and vectorial metrics

  Let $\mathbb{L} = (L,\leq,\max,\min )$ be a lattice with the join max{x, y}, and the meet min{x, y} on a set L ⊂ [0, ∞) which has a fixed number a as the greatest element and is closed under negation, i.e., for any x ∈ L, one has $\overline{x} = a - x \in L$.

  The scalar metric d on L is defined, for x ≠ y, by

  $$\displaystyle{d(x,y) =\max \{\min \{ x,\overline{y}\},\min \{\overline{x},y\}\}.}$$

  The scalar metric d ^∗ on L ^∗ = L ∪{∗}, ∗ ∉ L, is defined, for x ≠ y, by

  $$\displaystyle{d^{{\ast}}(x,y) = \left \{\begin{array}{ccc} d(x,y),&\mbox{ if }& x,y \in L,\\ \max \{x, \overline{x } \}, &\mbox{ if } & y = {\ast}, x\neq {\ast}, \\ \max \{y,\overline{y}\}, &\mbox{ if }&x = {\ast},y\neq {\ast}.\end{array} \right.}$$

  Given a norm | | . | | on $\mathbb{R}^{n}$, n ≥ 2, the vectorial metric on L ⁿ is defined by

  $$\displaystyle{\vert \vert (d(x_{1},y_{1}),\ldots,d(x_{n},y_{n}))\vert \vert,}$$

  and the vectorial metric on (L ^∗)ⁿ is defined by

  $$\displaystyle{\vert \vert (d^{{\ast}}(x_{ 1},y_{1}),\ldots,d^{{\ast}}(x_{ n},y_{n}))\vert \vert.}$$

  The vectorial metric on L ₂ ⁿ = { 0, 1}ⁿ with l ₁ -norm on $\mathbb{R}^{n}$ is the Fréchet–Nikodym–Aronszyan distance. The vectorial metric on $L_{m}^{n} =\{ 0, \frac{1} {m-1},\ldots, \frac{m-2} {m-1},1\}^{n}$ with l ₁ -norm on $\mathbb{R}^{n}$ is the Sgarro m-valued metric. The vectorial metric on [0, 1]ⁿ with l ₁ -norm on $\mathbb{R}^{n}$ is the Sgarro fuzzy metric.

  If L is L _m or [0, 1], and $x = (x_{1},\ldots,x_{n},x_{n+1},\ldots,x_{n+r})$, $y = (y_{1},\ldots,y_{n},{\ast},\ldots,{\ast})$, where ∗ stands in r places, then the vectorial metric between x and y is the Sgarro metric (see, for example, [CSY01]).

• Metrics on Riesz space

  A Riesz space (or vector lattice) is a partially ordered vector space (V _Ri , ⪯) in which the following conditions hold:

  • 1. The vector space structure and the partial order structure are compatible: x⪯ y implies $x + z\preceq y + z$, and x ≻ 0, $\lambda \in \mathbb{R},\lambda> 0$ implies λ x ≻ 0;

  • 2. For any two elements x, y ∈ V _Ri there exists the join x ∨ y ∈ V _Ri (in particular, the join and the meet of any finite set of elements from V _Ri exist).

  The Riesz norm metric is a norm metric on V _Ri defined by

  $$\displaystyle{\vert \vert x - y\vert \vert _{Ri},}$$

  where | | . | |  _Ri is a Riesz norm, i.e., a norm on V _Ri such that, for any x, y ∈ V _Ri , the inequality | x | ≤ | y | , where $\vert x\vert = (-x) \vee (x)$, implies | | x | |  _Ri  ≤ | | y | |  _Ri .

  The space (V _Ri , | | . | |  _Ri ) is called a normed Riesz space. In the case of completeness it is called a Banach lattice. All Riesz norms on a Banach lattice are equivalent.

  An element $e \in V _{Ri}^{+} =\{ x \in V _{Ri}: x \succ 0\}$ is called a strong unit of V _Ri if for each x ∈ V _Ri there exists $\lambda \in \mathbb{R}$ such that | x | ⪯ λ e. If a Riesz space V _Ri has a strong unit e, then $\vert \vert x\vert \vert =\inf \{\lambda \in \mathbb{R}: \vert x\vert \preceq \lambda e\}$ is a Riesz norm, and one obtains on V _Ri a Riesz norm metric

  $$\displaystyle{\inf \{\lambda \in \mathbb{R}: \vert x - y\vert \preceq \lambda e\}.}$$

  A weak unit of V _Ri is an element e of V _Ri ⁺ such that e ∧ | x | = 0 implies x = 0. A Riesz space V _Ri is called Archimedean if, for any two x, y ∈ V _Ri ⁺, there exists a natural number n, such that nx ⪯ y. The uniform metric on an Archimedean Riesz space with a weak unit e is defined by

  $$\displaystyle{\inf \{\lambda \in \mathbb{R}: \vert x - y\vert \wedge e\preceq \lambda e\}.}$$

• Machida metric

  For a fixed integer k ≥ 2 and the set $V _{k} =\{ 0,1,\ldots,k - 1\}$, let O _k ⁽ⁿ⁾ be the set of all n-ary functions from (V _k )ⁿ into V _k and $O_{k} = \cup _{n=1}^{\infty }O_{k}^{(n)}$. Let Pr _k be the set of all projections p r _i ⁿ over V _k , where $pr_{i}^{n}(x_{1},\ldots,x_{i},\ldots,x_{n}) = x_{i}$ for any $x_{1},\ldots,x_{n} \in V _{k}$.

  A clone over V _k is a subset C of O _k containing Pr _k and closed under (functional) composition. The set L _k of all clones over V _k is a lattice. The Post lattice L ₂ defined over Boolean functions, is countable but any L _k with k ≥ 3 is not. For n ≥ 1 and a clone C ∈ L _k , let C ⁽ⁿ⁾ denote n-slice C ∩ O _k ⁽ⁿ⁾.

  For any two clones C ₁, C ₂ ∈ L _k , Machida, 1998, defined the distance to be 0 if C ₁ = C ₂ and (min{n: C ₁ ⁽ⁿ⁾ ≠ C ₂ ⁽ⁿ⁾})⁻¹, otherwise. The lattice L _k of clones with this distance is a compact ultrametric space. Cf. Baire metric in Chap. 11.