Congruences on the lattice of pseudovarieties of finite semigroups.

*(English)*Zbl 0885.20036The 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)].

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)].

Reviewer: M.V.Volkov (Ekaterinburg)

##### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

08B15 | Lattices of varieties |

08A30 | Subalgebras, congruence relations |

08C15 | Quasivarieties |