Suppose \({ ( A , f ) }\) and \({ ( B , g ) }\) are semigroups. A function \({ h }\) from \({ A }\) to \({ B }\) is called a semigroup homomorphism if it has the property that \[ g ( h ( x ) , h ( y ) ) = h ( f ( x , y ) ) \] for all \({ x ∈ A }\) and \({ y ∈ A }\).
A semigroup homomorphism need not be a bijective function but if it is then its inverse function is also a semigroup homomorphism. In this case it’s called a semigroup isomorphism and the two semigroups are called isomorphic. In the particular case where both semigroups are the same the isomorphisms are called automorphisms.
If \({ B }\) is a subsemigroup of \({ A }\) and \({ g }\) is the restriction of \({ f }\) to \({ B }\) then the inclusion function is a semigroup homomorphism.
For a less trivial example, consider the natural numbers \({ N }\), with maximum as the operation, and the bicyclic semigroup \({ N ^ 2 }\) considered earlier. Then the function \({ h }\) defined by \[ h ( x ) = ( x , x ) \] is a semigroup homomorphism, since you can easily check that if \({ g }\) is the operation defined earlier, \[ g ( ( a , b ) , ( c , d ) ) = ( a + c - \min ( b , c ) , b + d - \min ( b , c ) ) , \] then \[ g ( ( x , x ) , ( y , y ) ) = ( \max ( x , y ) , \max ( x , y ) ) . \]
Suppose \({ ( A , f ) }\) and \({ ( B , g ) }\) are monoids. A function \({ h }\) from \({ A }\) to \({ B }\) is called a monoid homomorphism if it is a semigroup homomorphism and \({ h ( i ) = j }\), where \({ i }\) is the identity element of \({ ( A , f ) }\) and \({ j }\) is the identity element of \({ ( B , g ) }\).
A monoid homomorphism need not be a bijective function but if it is then its inverse function is also a monoid homomorphism. In this case it’s called a monoid isomorphism and the two monoids are called isomorphic. In the particular case where both monoids are the same the isomorphisms are called automorphisms.
The inclusion of submonoid in a monoid is a monoid homomorphism. The semigroup homomorphism from the natural numbers to the bicyclic monoid considered above is a monoid homomorphism since we’ve already seen that it’s a semigroup homeomorphism and we have \({ h ( 0 ) = ( 0 , 0 ) }\).
Another example of a monoid homomorphism is the length function on lists of items in a given set. This is a homomorphism from the set of lists, with the operation of concatenation, to the set of natural numbers, with the addition operation.
Suppose \({ ( A , f ) }\) and \({ ( B , g ) }\) are groups. A function \({ h }\) from \({ A }\) to \({ B }\) is called a group homomorphism if it is a monoid homomorphism. One could add the condition that \({ h }\) takes inverses to inverses but that’s redundant.
A group homomorphism need not be a bijective function but if it is then its inverse function is also a group homomorphism. In this case it’s called a group isomorphism and the two groups are called isomorphic. In the particular case where both groups are the same the isomorphisms are called automorphisms.
Note that the sets of semigroup automorphisms of a semigroup, monoid automorphisms of a monoid and group automorphisms of a group are all groups.