Generalized bicyclic semigroups and Jones semigroups
A classic result of Anderson is that if a simple, but not completely simple, semigroup \(S\) contains an idempotent, then it contains a copy of the bicyclic monoid \(B=\langle a,b\mid ab=1\rangle\). The reviewer [Proc. R. Soc. Edinb., Sect. A 106, 11-24 (1987; Zbl 0626.20047)] showed that if such a semigroup is idempotent-free and Green's relation \(\mathcal D\) is nontrivial, then it contains a copy of the semigroup \(C=\langle a,b\mid a^2b=a,\;ab^2=b\rangle\) or of \(A=\langle a,b\mid a^2b=a\rangle\) or its dual, each of which clearly has \(B\) as an image. Here the authors generalize the constructions of \(B\) and \(A\) as follows. For a positive integer \(n\), the `generalized bicyclic semigroup' \(B_n=\langle a,b\mid a^nb=1\rangle\); the `Jones semigroup' \(A_n=\langle a,b\mid a^{n+1}b=a\rangle\). The word problems are solved for these semigroups and used to study their properties. For instance, each \(B_n\) is bisimple and, for \(n>1\), right inverse but not inverse. The semigroups \(A_n\) inherit many of the properties of \(A=A_1\): each is the extension of an \(\mathcal L\)-trivial, simple, idempotent-free kernel by an infinite cyclic semigroup and has \(C\) as its minimum idempotent-free quotient. For distinct \(m,n\), \(A_m\) and \(A_n\) are mutually embeddable but nonisomorphic.
scientific article; zbMATH DE number 1748008