≤ t and st, then s = t. Similarly, if st. On E(S), the natural partial order becomes: so, since the idempotents form a semilattice under the product operation, products on E(S) give least upper bounds with respect to ≤. If E(S) is finite and forms a chain (i.e., E(S) is totally ordered by ≤), then S is a union of groups. If E(S) is an infinite chain it is possible to obtain an analogous result under additional hypotheses on S and E(S). Homomorphisms and representations of inverse semigroups A homomorphism (or morphism) of inverse semigroups is defined in exactly the same way as for any other semigroup: for inverse semigroups S and T, a function θ from S to T is a morphism if (sθ)(tθ) = (st)θ, for all s,t in S. The definition of a morphism of inverse semigroups could be augmented by including the condition (sθ)−1 = s−1θ, however, there is no need to do so, since this property follows from the above definition, via the following theorem: Theorem. The homomorphic image of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the image of that element. One of the earliest results proved about inverse semigroups was the Wagner–Preston Theorem, which is an analogue of Cayley's Theorem for groups: Wagner–Preston Theorem. If S is an inverse semigroup, then the function φ from S to , given by dom (aφ) = Sa−1 and x(aφ) = xa is a faithful representation of S. Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup, and with image closed under the inverse operation on partial bijections. Conversely, any subsemigroup of the symmetric inverse semigroup closed under the inverse operation is an inverse semigroup. Hence a semigroup S is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if S is an inverse semigroup. Congruences on inverse semigroups Congruences are defined on inverse semigroups in exactly the same way as for any other semigroup: a congruence ρ is an equivalence relation that is compatible with semigroup multiplication, i.e., Of particular interest is the relation , defined on an inverse semigroup S by there exists a with It can be shown that σ is a congruence and, in fact, it is a group congruence, meaning that the factor semigroup S/σ is a group. In the