On varieties of combinatorial inverse semigroups. II (Q1184168)

This is the continuation of the author's part I [ibid. 43, 305-330 (1991; Zbl 0743.20057)]. Relying on techniques and results introduced in the preceding paper, it is established that the variety of inverse semigroups \(\langle B_ 2^ 1\rangle\) generated by Perkins' semigroup \(B_ 2^ 1\) has no covers in the lattice of all varieties of combinatorial inverse semigroups. As a corollary of this remarkable result, the author proves that the inverse semigroup \(B_ 2^ 1\) has no irreducible basis of identities. This answers two questions proposed by \textit{E. I. Klejman} [Sib. Mat. Zh. 20, 760-777 (1979; Zbl 0417.20051)]. A key tool in the proof of the main result is a structural analysis of boundary combinatorial inverse semigroups (which are proved to be finite): completely semisimple combinatorial inverse semigroups not in \(\langle B_ 2^ 1\rangle\) all of whose proper inverse subsemigroups and proper quotients lie in \(\langle B_ 2^ 1\rangle\).