Minimal conditions on Clifford semigroup congruences

A partial group as defined in [3] is a semigroup S which satisfies the following axioms. (i) For every x ∈ S, there exists a (necessarily unique) element ex ∈ S, called the partial identity of x such that exx =xex =x and if yx =xy =x then ex y = yex = ex. (ii) For every x ∈ S, there exists a (necessarily unique) element x−1 ∈ S, called the partial inverse of x such that xx−1 = x−1x = ex and exx−1 = x−1ex = x−1. (iii) The operation x → ex is a homomorphism from S into S, that is, exy = exey for all x, y ∈ S, and the operation x → x−1 is an antihomomorphism, that is, (xy)−1 = y−1x−1 for all x, y ∈ S. Consequently, a partial group is precisely a Clifford semigroup, that is, a regular semigroup with central idempotents, and this is characterized by Clifford structure theorem (see [4, Chapter IV, Theorem 2.1] or [5, Chapter II, Theorem 2]) as a (strong) semilattice of groups. Thus, in particular, a partial group S may be viewed as a strong semilattice of groups S= [E(S);Se,φe, f ], where Se is the maximal subgroup of S with identity e (e ∈ E(S)) and for e ≥ f in E(S), φe, f is the homomorphism of groups Se → S f , x → x f . Here E(S) is the semilattice (e ≥ f if and only if e f = f ) of idempotents (partial identities) in S. Let S be a partial group. A subpartial group of S is a subsemigroup of S closed under the unary operations of S. A subpartial group of S is wide (or full) if it contains E(S). A normal subpartial group of S is a wide subpartial group K of S such that x−1Kx ⊂ K for all x ∈ S. This notion is standard in the literature, and we refer in particular to [2] for the following consequences.