Power exponents of aperiodic pseudovarieties
For a given algebraic structure \(A\) the power set \({\mathcal P}(A)\) carries a natural structure of the same type. For a pseudovariety \(V\) let \({\mathcal P}V\) denote the pseudovariety generated by all \({\mathcal P}(A)\) with \(A\in V\). The exponent of \(V\) is the least integer \(n\geq 0\) such that \({\mathcal P}^nV\) is the pseudovariety \(S\) of all finite semigroups. It was shown [\textit{J. Almeida}, Semigroup Forum 33, 375-390 (1986; Zbl 0588.20034)] that the iteration of the power operator on pseudovarieties of semigroups stabilizes at the end of three steps. Moreover, for non-permutative pseudovarieties, the stabilization occurs at the pseudovariety \(S\). The present paper shows how to compute the exponent of a non-permutative pseudovariety of aperiodic semigroups. Let \(Y\) be the semigroup \(\langle e,s,f:e^2=e,\;f^2=f,\;esf=s,\;ef=fe=0\rangle\), \(Q\) be the semigroup \(\langle e,s,t:e^2=e,\;es=s,\;te=t,\;se=et=0\rangle\), \(D\) be the five-element semigroup which coincides with \(D_2\) from \textit{A. V. Tishchenko} and \textit{M. V. Volkov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1995, No. 1(392), 91-99 (1995; Zbl 0846.20063)], \(D^\rho\) be its dual, \(T=F_3(LeSNB)\) be the free 3-generated semigroup of the pseudovariety of all finite left seminormal bands and \(T^\rho\) be its dual. Let \(V\) be a non-permutative pseudovariety of aperiodic semigroups. Then (a) \(V\) does not have exponent \(0\); (b) \(V\) has exponent 1 iff the five-element Brandt semigroup \(B_2\) belongs to \(V\); (c) \(V\) has exponent 2 iff \(B_2\not\in V\) and at least one of the semigroups \(Y\), \(Q\), \(D\), \(D^\rho\), \(T\), \(T^\rho\) belongs to \(V\); (d) \(V\) has exponent 3 iff none of the semigroups in (c) belongs to \(V\). In particular, the exponent of a decidable non-permutative aperiodic pseudovariety is computable.
scientific article; zbMATH DE number 1330649