Congruences on semigroups with the congruence extension property

Abstract

A congruence ρT on a subsemigroup T of S extends to the semigroup S, if there exists a congruence ρ on S such that the restriction ρ |T of ρ on T is ρT. A semigroup S has the congruence extension property, CEP, if each congruence on each subsemigroup extends to S.

Let S be a semigroup. Then S is called a GV-semigroup if S is π-regular and every regular element of S belongs to a subgroup(see[2]). For a congruence ρ on a semigroup S, the kernel kerρ of ρ is defined by {a ϵ S | a ρ e for some e ϵ E(SD)} and the trace trρ of ρ is defined by ρ | E(S). If S has CEP then by [14] S is a GV-semigroup. Thus we actually have kerρ = {a ϵ S | a ρ a ⁰}. It is known in [2] that a GV-semigroup S is a semilattice of nil-extensions of completely simple semigroups and the least semilattice congruence on S is

.

It is shown in [14] that a semigroup with CEP is a semilattice of nil-extensions of rectangular groups. The construction of semigroups with C'EP is given in [15].

In this paper, we first investigate congruences on several kinds of semigroups,such as nil-semigroups. π-groups and semigroups which are nil-extensions of rectangular groups. Then, we characterize congruences on a semigroup S = ∪{S _α | α ϵ Y} with CEP by a congruence aggregate approach. It is shown that a ρ b if and only if a ⁰ trρ b⁰ and ab ^-1 ϵ kerρ for a, b ϵ (RegS)ρ. Since D ^* ϵ C(S), we obtain the congruence pair (ρ ⋂ D ^*, ρ ⋁ D ^*), which is an analogue of (ρ ⋂ D, ρ ⋁ D) by Petrich [12]. In a sequel, we prove that a congruence ρ on a semigroup with CEP is determined uniquely by the triple (η, π, δ). In addition, we characterize congruences on semigroups with CEP by a kernel-trace approach. This method for congruences on semigroups is similar to Pet,rich [12] and [13].