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On free algebras in varieties generated by iterated semidirect products of semilattices. (English) Zbl 1269.20052

The free algebra of a variety \(V\) over an \(n\)-element set will be denoted \(\mathbf F_n(V)\). If \(V\) is locally finite, then the sequence \(f_n(V)=|\mathbf F_n(V)|\), \(n\geq 1\), consists of positive integers. It is called the ‘free spectrum’ of \({V}\). The authors note in introduction that if an algebra \(\mathbf A\) is finite, then the free spectrum of \(\mathbf A\), \(f_n(\mathbf A)\), is just the free spectrum of the variety that \(\mathbf A\) generates.
It is known from universal algebra, \(f_n(\mathbf A)\) is in fact the number of all \(n\)-ary operations on \(A\), the carrier set of \(\mathbf A\), induced by terms in the signature of \(\mathbf A\). These operations are called the ‘term operations’ of \(\mathbf A\). An \(n\)-ary term operation of \(\mathbf A\) is called ‘essentially \(n\)-ary’ if it depends on all of its variables, and \(p_n(\mathbf A)\) counts all such operations. In 1999 there was shown that the free spectrum of a general finite algebra \(\mathbf A\) is to a great deal governed by the free spectrum of an associated monoid, called the ‘twin monoid’ of \(\mathbf A\) [K. A. Kearnes, Algebra Univers. 42, No. 3, 165-182 (1999; Zbl 0978.08006)].
If \(\mathcal{SL}\) denotes the variety of all semilattices, we define a sequence of varieties \(\mathcal{SL}^t\), \(t\geq 1\), by \(\mathcal{SL}^{i+1}=\mathcal{SL}*\mathcal{SL}^i\) for all \(i\geq 1\). These varieties (and the corresponding pseudovarieties of finite semigroups, obtained by taking their finite members), generated by \(t\) times iterated semidirect products of semilattices, were thoroughly studied by J. Almeida [J. Algebra 142, No. 1, 239-254 (1991; Zbl 0743.20056)].
In this paper the authors supply an alternative solution of word problems for their free algebras, which will allow them to construct systems of normal forms of elements of these free algebras and calculate the free spectra and \(p_n\)-sequences of varieties of the form \(\mathcal{SL}^i\). The main result of the paper is the following. For each \(t\geq 1\) both \(\log f_n(t)\) and \(\log p_n(t)\) belong to the asymptotic class \(\mathcal O(n^t)\).

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
08B20 Free algebras
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References:

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