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A '''monoid homomorphism''' from a monoid <math>\underline{X}_1 = (X_1, *_1, e_1)\!</math> to a monoid <math>\underline{X}_2 = (X_2, *_2, e_2)\!</math> is a mapping between the underlying sets, <math>h : X_1 \to X_2,\!</math> that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements.  This means that the map <math>h\!</math> is a semigroup homomorphism from <math>\underline{X}_1\!</math> to <math>\underline{X}_2,\!</math> where these are considered as semigroups, but with the extra condition that <math>h\!</math> takes <math>e_1\!</math> to <math>e_2.\!</math>

A '''monoid homomorphism''' from a monoid <math>\underline{X}_1 = (X_1, *_1, e_1)\!</math> to a monoid <math>\underline{X}_2 = (X_2, *_2, e_2)\!</math> is a mapping between the underlying sets, <math>h : X_1 \to X_2,\!</math> that preserves the structure appropriate to monoids, namely, the LOCs and the identity elements.  This means that the map <math>h\!</math> is a semigroup homomorphism from <math>\underline{X}_1\!</math> to <math>\underline{X}_2,\!</math> where these are considered as semigroups, but with the extra condition that <math>h\!</math> takes <math>e_1\!</math> to <math>e_2.\!</math>

A '''group homomorphism''' from a group <math>\underline{X}_1 = (X_1, *_1, e_1)\!</math> to a group <math>\underline{X}_2 = (X_2, *_2, e_2)\!</math> is a mapping between the underlying sets, <math>h : X_1 \to X_2,\!</math> that preserves the structure appropriate to groups, namely, the LOCs, the identity elements, and the inverse elements.  This means that the map <math>h\!</math> is a monoid homomorphism from <math>X_1\!</math> to <math>X_2,\!</math> where these are viewed as monoids, with the extra condition that <math>h(x^{-1}) = h(x)^{-1}\!</math> for all <math>x \in X_1.\!</math> As it happens, the inverse elements are automatically preserved if the LOCs and the identity elements are, so a monoid homomorphism suffices to constitute a group homomorphism for a monoid that is also a group.  To see why this is so, consider the following chain of equalities:

A "group homomorphism" from a group X1 = <X1, *1, e1> to a group X2 = <X2, *2, e2> is a mapping between the underlying sets, h : X1 >X2, that preserves the structure appropriate to groups, namely, the LOCs, the identity elements, and the inverse elements.  This means that the map h is a monoid homomorphism from X1 to X2, where these are viewed as monoids, with the extra condition that h(x 1) = h(x) 1 for all x C X1.  As it happens, the inverse elements are automatically preserved if the LOCs and the identity elements are, so a monoid homomorphism suffices to constitute a group homomorphism for a monoid that is also a group.  To see why this is so, consider the following chain of equalities:

h(x) *2 h(x 1) = h(x *1 x 1) = h(e1) = e2.

| <math>h(x) *_2 h(x^{-1}) ~=~ h(x *_1 x^{-1}) ~=~ h(e_1) ~=~ e_2.\!</math>

An "isomorphism" is a homomorphism that is one to one and onto, or bijective.  Systems that have an isomorphism between them are called "isomorphic" to each other and belong to the same "isomorphism class".  From an abstract point of view, isomorphic systems are tantamount to the same mathematical object, differing at most in their manner of presentation and the details of their representation.  Usually these differences are regarded as purely notational, a mere change of names.  Thus, they are seen as accidental or accessory features of the object, corresponding to different ways of grasping the objective structure that is the main interest of the study but not considered as essential parts of its ultimate constitution or even necessary to its final comprehension.

An "isomorphism" is a homomorphism that is one to one and onto, or bijective.  Systems that have an isomorphism between them are called "isomorphic" to each other and belong to the same "isomorphism class".  From an abstract point of view, isomorphic systems are tantamount to the same mathematical object, differing at most in their manner of presentation and the details of their representation.  Usually these differences are regarded as purely notational, a mere change of names.  Thus, they are seen as accidental or accessory features of the object, corresponding to different ways of grasping the objective structure that is the main interest of the study but not considered as essential parts of its ultimate constitution or even necessary to its final comprehension.