Right cancellative and left ample monoids: quasivarieties and proper covers
Left ample monoids (called left type A in earlier papers by \textit{J.~Fountain} who introduced this notion [in Q. J. Math., Oxf. II. Ser. 28, 285-300 (1977; Zbl 0377.20051)]) can be defined as monoids with commuting idempotents and an additional unary operation \(x\mapsto x^+\) verifying the laws \((x^+)^+=x^+\), \(x^+x=x\), \(xy=zy\Rightarrow xy^+=zy^+\), \(x=x^2\Rightarrow yx=(yx)^+y\). Such monoids have proved to be related to right cancellative monoids in a way very similar to the way inverse semigroups are related to groups. In particular, the notion of a proper cover of a left ample monoid over a given quasivariety \(\mathcal V\) of right cancellative monoids naturally extends the well known notion of an \(E\)-unitary cover of an inverse semigroup over a given variety of groups. Studying proper covers, the author proves the following result analogous to the one obtained by \textit{M.~Petrich} and \textit{N.~R.~Reilly} [Acta Sci. Math. 46, 59-72 (1983; Zbl 0537.20032)] and \textit{F.~Pastijn} [Semigroup Forum 24, 87-88 (1982; Zbl 0503.20023)] in the inverse semigroup setting: if \(\mathcal V\) can be defined within the quasivariety of all right cancellative monoids by a set of identities of type \((2,0)\) then the class \(\widehat{\mathcal V}\) of all left ample monoids having a proper cover in \(\mathcal V\) can be defined by a set of identities of type \((2,1,0)\) within the quasivariety of all left ample monoids, namely, by the set \(\{u^+v=v^+u\mid u=v\) is an identity in \(\mathcal V\}\) (Theorem~5.3). The free objects in \(\widehat{\mathcal V}\) are then characterized as certain graph extensions of the free objects in \(\mathcal V\) (Theorem 5.4) -- the inverse forerunner of this result is due to \textit{S.~W.~Margolis} and \textit{J.~C.~Meakin} [J. Pure Appl. Algebra 58, No.~1, 45-76 (1989; Zbl 0676.20037)].
scientific article; zbMATH DE number 1471821
scientific article