Weakly reductive semigroups with atomistic congruence lattices

Abstract

The structure of semigroups with atomistic congruence lattices (that is, each congruence is the supremum of the atoms it contains) is studied. For the weakly reductive case the problem of describing the structure of such semigroups is solved up to simple and congruence free semigroups, respectively. As applications, all commutative, finite, completely semisimple semigroups, respectively, with atomistic congruence lattices are described.

Semigroups with atomistic congruence lattices

“…A straightforward verification shows that S is a semigroup. This construction also appears in [1] and [2], a similar one in [4]. By the latter paper it follows that each ideal of such a semigroup is a retract.…”

“…e X for which «(M, U) < n. If x T Q y then nothing has to be proved. In other words, if S -(X; I a , f a ") either has a weakly reductive kernel or no kernel at all then S is weakly reductive and we may apply the results of [2] . We still have to identify those inflations of S which have atomistic congruence lattices.…”

“…Suppose each homomorphic image of the semigroup S has an atomistic congruence lattice. If S has no kernel then S is of type (2) or an inflation thereof with trivial inflation function. Suppose that S has a kernel / .…”

“…In Section 2 we obtain necessary and suffi- [2] Semigroups with atomistic congruence lattices 89 cient conditions for a semigroup S to have an atomistic congruence lattice. In section 3 we study a complete congruence D on the congruence lattice of 5 .…”

Generalised retract semigroups

“…Furthermore, if X has a least element /x then by definition I* is closed under multiplication and thus it is a simple semigroup. Tully semigroups appear in different contexts in semigroup literature (see [1,2] and [3]).…”