PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

Abstract

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α → α + , where α + is the identity map on the domain of α. We investigate the construction of 'actions' from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.2000 Mathematics subject classification: primary 20M30; secondary 20M18.

Partial cohomology of groups

“…This question was first considered in the PhD Thesis [1] (see also [2]) and independently in [45] and [39] for partial group actions, with subsequent developments in [3,12,17,18,21,22,24,32,33,40,44]. More generally the problem was investigated for partial semigroup actions in [37,38,41,43], for partial groupoid actions in [10,11,36] and around partial Hopf (co)actions in [5,6,7,8,14,15,16].…”

Domain and Range Operations in Semigroups and Rings

“…The D-semiadequate semigroups embeddable in X for some X can be specified by one additional equation (namely, aD b = D ab a), thereby giving the variety of left restriction semigroups, considered by a great many authors and under many different names; see [34] and [36] where the axiomatization was first shown to be complete, [19] and [26] where this was re-discovered, as well as [12] and [13], where they arise as weakly left E-ample semigroups. The axioms for those Dsemiadequate DR-semigroups embeddable in X equipped with both domain and range operations were provided by [32], following a correction to a result in [34].…”

D-semigroups and constellations

“…This representation theorem was first given by Trokhimenko in [20]. Left restriction semigroups are also known as weakly type SL γ-semigroups in [1], left E-ample semigroups (a term first used in [5] and then again in papers including [8]), guarded semigroups (see [16]), and twisted LC-semigroups (see [11] and [12]). It is therefore natual to seek to extend Lawson's results, at least to left restriction semigroups.…”