lattice. An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group. Green's relations, a set of five equivalence relations that characterise the elements in terms of the principal ideals they generate, are important tools for analysing the ideals of a semigroup and related notions of structure. The subset with the property that every element commutes with any other element of the semigroup is called the center of the semigroup. The center of a semigroup is actually a subsemigroup. Homomorphisms and congruences A semigroup homomorphism is a function that preserves semigroup structure. A function between two semigroups is a homomorphism if the equation . holds for all elements a, b in S, i.e. the result is the same when performing the semigroup operation after or before applying the map f. A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there are semigroup homomorphisms which are not monoid homomorphisms, e.g. the canonical embedding of a semigroup without identity into . Conditions characterizing monoid homomorphisms are discussed further. Let be a semigroup homomorphism. The image of is also a semigroup. If is a monoid with an identity element , then is the identity element in the image of . If is also a monoid with an identity element and belongs to the image of , then , i.e. is a monoid homomorphism. Particularly, if is surjective, then it is a monoid homomorphism. Two semigroups S and T are said to be isomorphic if there exists a bijective semigroup homomorphism . Isomorphic semigroups have the same structure. A semigroup congruence is an equivalence relation that is compatible with the semigroup operation. That is, a subset that is an equivalence relation and and implies for every in S. Like any equivalence relation, a semigroup congruence induces congruence classes and the semigroup operation induces a binary operation on the congruence classes: Because is a congruence, the set of all congruence classes of forms a semigroup with , called the quotient semigroup or factor semigroup, and denoted . The mapping is a semigroup homomorphism, called the quotient map, canonical surjection or projection; if S is a monoid then quotient semigroup is a monoid with identity . Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. These results are nothing more