Morita invariants for semigroups with local units

Abstract

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.

Fair semigroups and Morita equivalence

“…In [7], several properties are shown to be invariant under strong Morita equivalence in the presence of factorisability or various kinds of local units. Example 7 shows that the properties of being a group, being a monoid, having local units or weak local units, and factorisability are not invariant under Morita equivalence in the class of all semigroups.…”

“…In recent years, a satisfactory theory of Morita equivalence has been developed for semigroups with local units, see [6][7][8]. The efficiency of this theory is largely due to the fact, established by Lawson [8], that in this case Morita equivalence is strong in the sense that it comes from a well-behaved Morita context.…”

Morita invariants for partially ordered semigroups with local units

“…being completely 0-simple or bisimple. More recently, Lawson [14], Laan [11], and Laan and Márki [12] have examined more invariants by using both Morita contexts and various new characterizations of Morita equivalence. For example, Lawson [14] implicitly shows that the property of being regular and locally inverse is a Morita invariant.…”

“…Recently, there has been a resurgence of interest in the Morita theory of semigroups [11][12][13][14]18]. The current work is part of an attempt (initiated in [20]) to generalize some of those latest results from semigroups to partially ordered semigroups, and belongs to a line of research trying to establish generalizations of various results of semigroup theory to partially ordered semigroups (e.g.…”

Characterizations of strong Morita equivalence for ordered semigroups with local units

“…A (po)semigroup S is said to have common (weak) local units (cf. [5]) if for any s, s ∈ S there exist e ∈ E(S) and f ∈ E(S) (e ∈ S and f ∈ S) such that es = s = sf and es = s = s f . We say that a posemigroup S has ordered (weak) local units if for all s, s ∈ S, s ≤ s , there exist e, e , f, f ∈ E(S) (e, e , f, f ∈ S) such that…”