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Next, the concept of a homomorphism or a ''structure-preserving map'' is specialized to the different kinds of structure of interest here.

Next, the concept of a homomorphism or ''structure-preserving map'' is specialized to the different kinds of structure of interest here.

A '''semigroup homomorphism''' from a semigroup <math>\underline{X}_1 = (X_1, *_1)\!</math> to a semigroup <math>\underline{X}_2 = (X_2, *_2)\!</math> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs.  This makes it a map <math>h : X_1 \to X_2\!</math> whose induced action on the LOCs is such that it takes every element of <math>*_1\!</math> to an element of <math>*_2.\!</math> That is:

A "semigroup homomorphism" from a semigroup X1 = <X1, *1> to a semigroup X2 = <X2, *2> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs.  This makes it a map h : X1 >X2 whose induced action on the LOCs is such that it takes every element of *1 to an element of *2.  That is:

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

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

A "monoid homomorphism" from a monoid X1 = <X1, *1, e1> to a monoid X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements.  This means that the map h is a semigroup homomorphism from X1 to X2, where these are considered as semigroups, but with the extra condition that h takes e1 to e2.

A "monoid homomorphism" from a monoid X1 = <X1, *1, e1> to a monoid X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements.  This means that the map h is a semigroup homomorphism from X1 to X2, where these are considered as semigroups, but with the extra condition that h takes e1 to e2.