Strong Morita equivalence for ordered semigroups with local units

On Morita equivalence of partially ordered semigroups with local units

“…The Morita theory for semigroups has seen some significant recent advancement by Lawson in [9], who provided a definition of Morita equivalence in terms of "closed" acts and showed that such a definition is equivalent to both Talwar's original definition ( see [11]) and, more importantly, to a number of other algebraic conditions. In a follow-up article to Lawson's (see [15], which is extended in [13]), we proved that Lawson's conditions except for the definition of Morita equivalence are still equivalent for partially ordered semigroups with local units. We defined what we called "strong Morita equivalence" (cf.…”

“…First, let us recall from [15] the ordered analogue of Theorem 1.1 of [9], showing the equivalence of the algebraic characterizations.…”

Characterizations of strong Morita equivalence for ordered semigroups with local units

“…Roughly half of it has been proved in [12] and the subsequent sections of the article are devoted to examining each of the three remaining conditions in detail and establishing their equivalence with the others. The following is an important observation about the structure of strongly Morita equivalent posemigroups, namely that they have the same local structure.…”

“…It has seen quite a bit of recent development by Lawson ([7]) and Laan and Márki ( [6]), who present a number of alternative characterizations of Morita equivalent semigroups. Our work considers how those results can be applied to the Morita theory of posemigroups and is a sequel to [12]. We use the main result of [12] to examine how strong Morita equivalence is connected to the existence of a strict local isomorphism between a posemigroup T and several posemigroups constructed from a given posemigroup S: the Cauchy completion C(S) viewed as a posemigroup, a Rees matrix posemigroup over S and a Morita posemigroup over S. Along the way, we describe the Rees matrix posemigrops corresponding to unipotent pomonoids and show that Morita posemigroups over S are always strongly Morita equivalent to S. Most of the results are refinements of the Morita theory of semigroups with (various kinds of) local units as found in [6], [7] and [10].…”

Morita invariants for partially ordered semigroups with local units

“…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. [6,16,17], an overview in [5]).…”