On free spectra of a class of finite inverse monoids. (Q2840477)

The free spectrum of a finite algebra is the sequence of cardinalities of the free members of rank \(n\), \(n=1,2,\ldots\), in the variety that it generates. Such an algebra has a doubly exponential free spectrum if its values are as `large as possible', that is, there is a positive real number such that, for all sufficiently large \(n\), the free algebra of rank \(n\) over \(A\) has cardinality at least \(2^{2^{cn}}\). Otherwise, it is sub-log-exponential.NEWLINENEWLINE \textit{S. Seif} [J. Pure Appl. Algebra 212, No. 5, 1162-1174 (2008; Zbl 1138.20049)] conjectured a simple necessary and sufficient condition that a finite monoid be sub-log-exponential. Any finite inverse monoid with nilpotent subgroups satisfies this necessary condition. The author provides further evidence for the conjecture by verifying it for a class of such monoids that arises from Clifford inverse algebras by taking semidirect products with their underlying semilattices via a `term-expressible' action; then validity of the conjecture holds for a broader class of finite inverse semigroups, as a consequence. (An inverse algebra, in the sense of \textit{J. Leech} [Proc. Lond. Math. Soc., III. Ser. 70, No. 1, 146-182 (1995; Zbl 0830.20086)], is an inverse monoid whose natural partial order is a semilattice, thereby extending its set of operations. It is Clifford if its underlying inverse monoid is a Clifford semigroup.)