Varieties of \(P\)-restriction semigroups. (English) Zbl 1303.20062
As the author writes in the introduction to the work [P. R. Jones, J. Pure Appl. Algebra 216, No. 3, 618-632 (2012; Zbl 1257.20058)] he has introduced the notion “\(P\)-restriction semigroups as a common generalization of the restriction semigroups (or weakly \(E\)-ample semigroups) and regular \(*\)-semigroups, defining them as bi-unary semigroups – semigroups with two additional unary operations, \(^+\) and \(^*\) – satisfying a set of simple identities. The projections in such a semigroup are the elements of the form \(x^+\) (or, equivalently, \(x^*\)). If \((S,\cdot,{^{-1}})\) is a regular \(*\)-semigroup and \(^+\) and \(^*\) are defined, respectively, by \(x^+=xx^{-1}\) and \(x^*=x^{-1}x\), then its bi-unary reduct is a \(P\)-restriction semigroup …. For any variety \(\mathbf V\) of regular \(*\)-semigroups, let \(C\mathbf V\) (resp. \(\mathbf PC\mathbf V\)) denote the variety of regular \(*\)-semigroups (\(P\)-restriction semigroups) whose projections generate a member of \(\mathbf V\). For instance, if \(\mathbf{Sl}\) is the variety of \(*\)-semilattices, \(C\mathbf V\) and \(\mathbf PC\mathbf V\) are the varieties of inverse semigroups and of restriction semigroups, respectively. The general question is then, “For which varieties \(\mathbf V\) of regular \(*\)-semigroups is it true that the variety of \(P\)-restriction semigroups generated by \(C\mathbf V\) is precisely \(\mathbf PC\mathbf V\)? It is shown that for a given variety \(\mathbf V\), a positive answer to this question is equivalent to the following one: For each nonempty set \(X\), the free \(P\)-restriction semigroup in \(\mathbf PC\mathbf V\), on \(X\), is isomorphic to the bi-unary subsemigroup of the free regular \(*\)-semigroup in \(C\mathbf V\), on \(X\), that is generated by \(X\).”
As the author writes in the abstract this “relationship between varieties of regular \(*\)-semigroups and varieties of \(P\)-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox \(*\)-semigroups and varieties of orthodox \(P\)-restriction semigroups, leading to concrete descriptions of the free orthodox \(P\)-restriction semigroups.” There are some other results in this direction.
As the author writes in the abstract this “relationship between varieties of regular \(*\)-semigroups and varieties of \(P\)-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox \(*\)-semigroups and varieties of orthodox \(P\)-restriction semigroups, leading to concrete descriptions of the free orthodox \(P\)-restriction semigroups.” There are some other results in this direction.
Reviewer: Aleksandr V. Tishchenko (Moskva)
MSC:
| 20M07 | Varieties and pseudovarieties of semigroups |
| 20M05 | Free semigroups, generators and relations, word problems |
| 08B15 | Lattices of varieties |
Citations:
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