than a particularization of the first isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study in string rewriting systems. A nuclear congruence on S is one which is the kernel of an endomorphism of S. A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered by inclusion, has a maximal element. By Zorn's lemma, this is equivalent to saying that the ascending chain condition holds: there is no infinite strictly ascending chain of congruences on S. Every ideal I of a semigroup induces a factor semigroup, the Rees factor semigroup, via the congruence ρ defined by if either , or both x and y are in I. Quotients and divisions The following notions introduce the idea that a semigroup is contained in another one. A semigroup T is a quotient of a semigroup S if there is a surjective semigroup morphism from S to T. For example, is a quotient of , using the morphism consisting of taking the remainder modulo 2 of an integer. A semigroup T divides a semigroup S, noted if T is a quotient of a subsemigroup S. In particular, subsemigroups of S divides T, while it is not necessarily the case that there are a quotient of S. Both of those relation are transitive. Structure of semigroups For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xn n ∈ Z+ }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent. A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each