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The next series of definitions develops the mathematical concepts of ''homomorphism'' and ''isomorphism'', special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively.  In very rough terms, a ''homomorphism'' is a ''structure-preserving mapping'' between systems, but only in the sense that it preserves some part or some aspect of the structure mapped, whereas an ''isomorphism'' is a correspondence that preserves all of the relevant structure.

The next series of definitions develops the mathematical concepts of ''homomorphism'' and ''isomorphism'', special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively.  In very rough terms, a ''homomorphism'' is a ''structure-preserving mapping'' between systems, but only in the sense that it preserves some part or some aspect of the structure mapped, whereas an ''isomorphism'' is a correspondence that preserves all of the relevant structure.

The '''induced action''' of a function <math>f : X\to Y\!</math> on the cartesian power <math>X^n\!</math> is the function <math>f' : X^n \to Y^n\!</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_n) ~=~ (fx_1, \ldots, fx_n).\!</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^n \to Y^n,\!</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_n) ~=~ (fx_1, \ldots, fx_n).\!</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^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^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~ (hx, hy, hz) \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:

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

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

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