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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^n\!</math> is a mapping between the underlying sets, <math>h : X \to Y,\!</math> whose induced action <math>h : X^n \to Y^n\!</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^n\!</math> is a mapping between the underlying sets, <math>h : X \to Y,\!</math> whose induced action <math>h : X^n \to Y^n\!</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 ~\mapsto~ 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 ~\mapsto~ 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 h : X1 >X2, or its tacit extension as a mapping h : X13 >X23, this implication yields the following:

<x, y, z> C *1  =>  <h(x), h(y), h(z)> C *2.

: <math>(x, y, z) \in *_1 ~\Rightarrow~ (hx, hy, hz) \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:

x *1 y = z.

: <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:

h(x) *2 h(y)  = h(z).

: <math>hx *_2 hy = hz.\!</math>

From these two equations one derives, by substituting x *1 y for z in h(z), a succinct formulation of the condition for a mapping h : X1 >X2 to be a relation homomorphism from a system <X1, *1> to a system <X2, *2>, expressed in the form of a "distributive law" or "linearity condition":

From these two equations one derives, by substituting x *1 y for z in h(z), a succinct formulation of the condition for a mapping h : X1 >X2 to be a relation homomorphism from a system <X1, *1> to a system <X2, *2>, expressed in the form of a "distributive law" or "linearity condition":