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Given a SOD <math>(X_i),\!</math> its cartesian product, notated as <math>\textstyle\prod_i (X_i)</math> or <math>\textstyle\prod_i X_i,</math> is defined as follows:

Given a SOD <math>(X_i),\!</math> its cartesian product, notated as <math>\textstyle\prod_i (X_i)</math> or <math>\textstyle\prod_i X_i,</math> is defined as follows:

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{| align="center" cellspacing="8" width="90%"

| <math>\prod_i (X_i) = \prod_i X_i = \{ (x_i) : x_i \in X_i \}.</math>

| <math>\prod_i (X_i) = \prod_i X_i = \{ (x_i) : x_i \in X_i \}.</math>

|}

|}

An '''inverse''' of an element <math>x\!</math> in a monoid <math>\underline{X} = (X, *, e)\!</math> is an element <math>y \in X\!</math> such that <math>x*y = e = y*x.\!</math>  An element that has an inverse in <math>\underline{X}\!</math> is said to be '''invertible''' (relative to <math>*\!</math> and <math>e\!</math>).  If <math>x\!</math> has an inverse in <math>\underline{X},\!</math> then it is unique to <math>x.\!</math>  To see this, suppose that <math>y'\!</math> is also an inverse of <math>x.\!</math>  Then it follows that:

An '''inverse''' of an element <math>x\!</math> in a monoid <math>\underline{X} = (X, *, e)\!</math> is an element <math>y \in X\!</math> such that <math>x*y = e = y*x.\!</math>  An element that has an inverse in <math>\underline{X}\!</math> is said to be '''invertible''' (relative to <math>*\!</math> and <math>e\!</math>).  If <math>x\!</math> has an inverse in <math>\underline{X},\!</math> then it is unique to <math>x.\!</math>  To see this, suppose that <math>y'\!</math> is also an inverse of <math>x.\!</math>  Then it follows that:

: <math>y' ~=~ y'*e ~=~ y'*(x*y) ~=~ (y'*x)*y ~=~ e*y ~=~ y.\!</math>

| <math>y' ~=~ y'*e ~=~ y'*(x*y) ~=~ (y'*x)*y ~=~ e*y ~=~ y.\!</math>

A '''group''' is a monoid all of whose elements are invertible.  That is, a group is a semigroup with a unit element in which every element has an inverse.  Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:

A '''group''' is a monoid all of whose elements are invertible.  That is, a group is a semigroup with a unit element in which every element has an inverse.  Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:

The '''induced action''' of a function <math>f : X\to Y\!</math> on the cartesian power <math>X^k\!</math> is the function <math>f' : X^k \to Y^k\!</math> defined by:

The '''induced action''' of a function <math>f : X\to Y\!</math> on the cartesian power <math>X^k\!</math> is the function <math>f' : X^k \to Y^k\!</math> defined by:

: <math>f'((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!</math>

| <math>f'((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!</math>

Usually, <math>f'\!</math> is regarded as the ''natural'', ''obvious'', ''tacit'', or ''trivial'' extension that <math>f : X \to Y\!</math> possesses in the space of functions <math>X^k \to Y^k,\!</math> and is thus allowed to go by the same name as <math>f.\!</math>  This convention, assumed by default, is expressed by the formula:

Usually, <math>f'\!</math> is regarded as the ''natural'', ''obvious'', ''tacit'', or ''trivial'' extension that <math>f : X \to Y\!</math> possesses in the space of functions <math>X^k \to Y^k,\!</math> and is thus allowed to go by the same name as <math>f.\!</math>  This convention, assumed by default, is expressed by the formula:

: <math>f((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!</math>

| <math>f((x_1, \ldots, x_k)) ~=~ (f(x_1), \ldots, f(x_k)).\!</math>

A '''relation homomorphism''' from a <math>k\!</math>-place relation <math>P \subseteq X^k\!</math> to a <math>k\!</math>-place relation <math>Q \subseteq Y^k\!</math> is a mapping between the underlying sets, <math>h : X \to Y,\!</math> whose induced action <math>h : X^k \to Y^k\!</math> preserves the indicated relations, taking every element of <math>P\!</math> to an element of <math>Q.\!</math>  In other words:

A '''relation homomorphism''' from a <math>k\!</math>-place relation <math>P \subseteq X^k\!</math> to a <math>k\!</math>-place relation <math>Q \subseteq Y^k\!</math> is a mapping between the underlying sets, <math>h : X \to Y,\!</math> whose induced action <math>h : X^k \to Y^k\!</math> preserves the indicated relations, taking every element of <math>P\!</math> to an element of <math>Q.\!</math>  In other words:

: <math>(x_1, \ldots, x_k) \in P ~\Rightarrow~ h((x_1, \ldots, x_k)) \in Q.\!</math>

| <math>(x_1, \ldots, x_k) \in P ~\Rightarrow~ h((x_1, \ldots, x_k)) \in Q.\!</math>

Applying this definition to the case of two binary operations, say <math>*_1\!</math> on <math>X_1\!</math> and <math>*_2\!</math> on <math>X_2,\!</math> which are special kinds of triadic relations, say <math>*_1 \subseteq X_1^3\!</math> and <math>*_2 \subseteq X_2^3,\!</math> one obtains:

Applying this definition to the case of two binary operations, say <math>*_1\!</math> on <math>X_1\!</math> and <math>*_2\!</math> on <math>X_2,\!</math> which are special kinds of triadic relations, say <math>*_1 \subseteq X_1^3\!</math> and <math>*_2 \subseteq X_2^3,\!</math> one obtains:

: <math>(x, y, z) \in *_1 ~\Rightarrow~ h((x, y, z)) \in *_2.\!</math>

| <math>(x, y, z) \in *_1 ~\Rightarrow~ h((x, y, z)) \in *_2.\!</math>

Under the induced action of <math>h : X_1 \to X_2,\!</math> or its tacit extension as a mapping <math>h : X_1^3 \to X_2^3,\!</math> this implication yields the following:

Under the induced action of <math>h : X_1 \to X_2,\!</math> or its tacit extension as a mapping <math>h : X_1^3 \to X_2^3,\!</math> this implication yields the following:

: <math>(x, y, z) \in *_1 ~\Rightarrow~ (h(x), h(y), h(z)) \in *_2.\!</math>

| <math>(x, y, z) \in *_1 ~\Rightarrow~ (h(x), h(y), h(z)) \in *_2.\!</math>

The left hand side of this implication is expressed more commonly as:

The left hand side of this implication is expressed more commonly as:

: <math>x *_1 y = z.\!</math>

| <math>x *_1 y = z.\!</math>

The right hand side of the implication is expressed more commonly as:

The right hand side of the implication is expressed more commonly as:

: <math>h(x) *_2 h(y) = h(z).\!</math>

| <math>h(x) *_2 h(y) = h(z).\!</math>

From these two equations one derives, by substituting <math>x *_1 y\!</math> for <math>z\!</math> in <math>h(z),\!</math> a succinct formulation of the condition for a mapping <math>h : X_1 \to X_2\!</math> to be a relation homomorphism from a system <math>(X_1, *_1)\!</math> to a system <math>X_2, *_2,\!</math> expressed in the form of a ''distributive law'' or ''linearity condition'':

From these two equations one derives, by substituting <math>x *_1 y\!</math> for <math>z\!</math> in <math>h(z),\!</math> a succinct formulation of the condition for a mapping <math>h : X_1 \to X_2\!</math> to be a relation homomorphism from a system <math>(X_1, *_1)\!</math> to a system <math>X_2, *_2,\!</math> expressed in the form of a ''distributive law'' or ''linearity condition'':

: <math>h(x *_1 y) ~=~ h(x) *_2 h(y).\!</math>

| <math>h(x *_1 y) ~=~ h(x) *_2 h(y).\!</math>

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