The bifree locally inverse semigroup on a set
A `locally inverse' semigroup is a regular semigroup \(S\) all of whose local submonoids (those of the form \(eSe\), for \(e \in E(S)\), its set of idempotents) are inverse. The class \({\mathcal L}{\mathcal I}\) of locally inverse semigroups is closed under products, homomorphic images and regular subsemigroups and as such forms an existence (or \(e\)-) variety. It was shown by \textit{Y. T. Yeh} [Int. J. Algebra Comput. 2, 471-484 (1992; Zbl 0765.20030)] that every \(e\)-variety \(\mathcal V\) of locally inverse semigroups possesses a `bifree object' \(BF{\mathcal V}(X)\) for each nonempty set \(X\), defined by the property that every `matched' mapping \(\phi\) from \(X \cup X'\) (where \(X'\) is a set of formal inverses for the elements of \(X\) and `matched' requires that \(x'\phi\) be an inverse of \(x \phi\) in \(S\) for every \(x\) in \(X\)) to a member \(S\) of \(\mathcal V\) extends uniquely to a morphism from \(BF{\mathcal V}(X)\) to \(S\). The purpose of the current paper is to describe \(BF{\mathcal L}{\mathcal I}(X)\). By a result of \textit{F. Pastijn} [Trans. Am. Math. Soc. 273, 631-655 (1982; Zbl 0512.20042)], every locally inverse semigroup is a homomorphic image of a regular subsemigroup of a ``semidirect product of a semilattice by a completely simple semigroup''. The construction of the latter generalizes both the Rees construction and the McAlister construction of `\(P\)-semigroups'. It follows that such semigroups generate \(\mathcal L \mathcal I\) as an \(e\)-variety and that \(BF{\mathcal L}{\mathcal I})(X)\) is embedded in a semigroup of this type. As a necessary and interesting preliminary, the author first provides a Rees representation for the bifree completely simple semigroup \(BF{\mathcal C}{\mathcal S}(X)\) on \(X\). The `semidirect product' of a certain free semilattice with this completely simple semigroup is then constructed; we denote it by \(P\), for convenience. To identify the requisite subsemigroup, a further notion is needed. Each locally inverse semigroup \(S\) carries a second binary operation \(*\), where \(x * y\) is the unique element in the sandwich set \(S(x'x,yy')\), where \(x'\), \(y'\) are any inverses of \(x\), \(y\), respectively. Thus it is natural to consider the free object \(F\) on \(X \cup X'\) in the class of all `binary semigroups', that is, algebras of type \(\langle 2,2\rangle\). A map \(\theta: X \cup X' \to P\) is defined. Finally, \(BF{\mathcal L}{\mathcal I}\) is the image of \(F\) in \(P\), under the natural extension of \(\theta\), that is, it is the binary subsemigroup of \(P\) generated by the image of \(X \cup X'\). While a determination of the actual elements of \(P\) that belong to \(BF{\mathcal L}{\mathcal I}\) is not obtained, the above description suffices to uncover many interesting properties of the latter, and solves the word problem there, in a certain sense. Various related results are also obtained. Reviewer's remarks: (1) The author follows Pastijn [loc. cit.] in his use of the term `semidirect product'; it is in fact not such a product, in any usual sense of the term. Some comments to this effect close the paper. However, this misleading term continues to appear in similar contexts. (2) In a sequel [``On the bifree locally inverse semigroup'' (to appear)], the author has rectified a weakness in the paper under review (the lack of a method for determining membership in \(BF{\mathcal L}{\mathcal I}(X)\) within \(P\)) by using a somewhat different approach to produce a set of canonical forms.
scientific article; zbMATH DE number 614247