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Congruences on the lattice of pseudovarieties of finite semigroups. (English) Zbl 0885.20036
The authors study two ways for constructing complete congruences on the lattice $$L(\mathcal F)$$ of pseudovarieties of finite semigroups. Such congruences are kernels of complete homomorphisms of $$L(\mathcal F)$$ which have the form $$\mathcal U\mapsto\mathcal U\cap\mathcal A$$ where $$\mathcal A$$ is a special class of finite semigroups. In the first approach (extending an idea previously applied by M. Petrich and N. R. Reilly [J. Aust. Math. Soc., Ser. A 49, No. 1, 1-23 (1990; Zbl 0708.20019)] to varieties of completely regular semigroups), $$\mathcal A$$ is a preimage class, that is, a class closed under finite direct products and homomorphic images and such that, for every surjective homomorphism $$\theta\colon S\to T$$ between finite semigroups $$S$$ and $$T\in\mathcal A$$, there is a subsemigroup $$R$$ of $$S$$ with $$R\in\mathcal A$$ and $$R\theta=T$$. The second approach follows the method used for varieties of completely regular semigroups by M. Petrich and N. R. Reilly [J. Algebra 134, No. 1, 1-27 (1990; Zbl 0706.20043)] and then for e-varieties of regular semigroups by N. R. Reilly and S. Zhang [J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051)]. Here $$\mathcal A$$ is the radical class of a radical congruence system, the latter being a family of congruences $$\{\kappa_S\}$$ indexed by finite semigroups which nicely behaves with respect to finite direct products, homomorphic images and subsemigroups and has the property that $$\kappa_{(S/\kappa_S)}$$ is the identity relation on $$S/\kappa_S$$ for each finite semigroup $$S$$. The radical class of such a radical congruence system $$\{\kappa_S\}$$ consists of all finite semigroups $$S$$ with $$\kappa_S$$ being the identity relation.
The authors present several concrete examples of complete congruences on the lattice $$L(\mathcal F)$$ constructed via either preimage or radical classes. The examples arising from radical classes are studied in more detail; in particular, the upper limits of the corresponding congruence classes of $$L(\mathcal F)$$ are characterized as certain Mal’cev products (Theorem 6.4).
As an important application, the authors demonstrate how to calculate the Krohn-Rhodes complexity $$Sc$$ of every finite semigroup $$S$$ in which the union of all subgroups forms a subsemigroup (Theorem 9.2(ii)). This extends a well known result by J. Rhodes [see Chapter 9 in M. A. Arbib, Algebraic theory of machines, languages and semigroups, Academic Press (1968; Zbl 0181.01501)]. They also show that if the subsemigroup $$C(S)$$ generated by all idempotents of a semigroup $$S$$ is completely regular then $$C(S)c\leq Sc\leq C(S)c+1$$ (Theorem 9.2(i)) thus generalizing a recent result by P. G. Trotter [J. Pure Appl. Algebra 105, No. 3, 319-328 (1995; Zbl 0845.20050)].

##### MSC:
 20M07 Varieties and pseudovarieties of semigroups 08B15 Lattices of varieties 08A30 Subalgebras, congruence relations 08C15 Quasivarieties
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