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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 ~\mapsto~ 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 or LOCs, say *1 on X1 and *2 on X2, which are special kinds of triadic relations, say *1 c X13 and *2 c X23, one obtains:

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

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

Under the induced action of h : X1 >X2, or its tacit extension as a mapping h : X13 >X23, 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: