NORMALLY ORDERED SEMIGROUPS
Vítor Hugo Fernandes · Glasgow Mathematical Journal · 2008
Abstract In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the classNOSof all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev productsNOS=EI $\petite m$ POIandNOS∩A=N $\petite m$ POI, whereA,EIandNdenote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, andPOIdenotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalitiesBG=EI $\petite m$ Ecom=N $\petite m$ Ecom, whereBGandEcomdenote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.