Francis Pastijn

Congruences on regular semigroups

Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation ker p, is the union of the idempotent eclasses. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp, trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((p V £)/£, ker p, (p V R)/R) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. For congruence relations p and 0, put pTz 0 [pTrr 0, pTo] if andonlyifpV: =0V: [pvR =0VR,trp=tr0]. ThenTz, Tr and T are complete congruences on the congruence lattice of S and T = Tl n T. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found in [10, Chapter III]. It was recognized by Feigenbaum [3] that every congruence p on a regular semigroup S is uniquely determined by its kernel, kerp, equal to elements tequivalent to idempotents, and its trace, trp, equal to the restriction of p to the set E(S) of idempotents of S. In the case of an inverse semigroup S, kerp and trp, as well as their mutual relationship, can be described abstractly by means of simple conditions on a subset of S and an equivalence on E(S) (see [10, Chapter III]). Following in the footsteps of Scheiblich, for orthordox and arbitrary regular semigroups, Feigenbaum [3] and lYotter [14] adopted the following approach: trp is characterized abstractly and to each such trp all matching kernels are described. This unbalances the symmetry of the kernel-trace approach by giving preference to the trace. Hence a balanced view relative to the kernel and the trace is evidently called for. The unqualified success of the kernel-trace approach for inverse semigroups, including its diverse ramifications, gave a certain hope that this may also turn out to be the case for regular semigroups. Judging by the complexity of regular semigroups and the attempts made for both orthodox and general regular semigroups, Received by the editors November 2, 1984. 1980 Mathematics Subject Classification. Primary 20M10; Secondary 08A30. (a)1986 American Mathematical Society 0002-9947/86

The lattice of completely regular semigroup varieties

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The Congruence Lattice of a Regular Semigroup

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Complete Congruences on Lattices of Varieties and of Pseudovarieties

A completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice £(CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on £(CR) yields a subdirect decomposition of £(CR). Using these results we show that £(CR) is arguesian. This confirms the (tacit) conjecture that £(CR) is modular. 1980 Mathematics subject classification {Amer. Math. Soc.) (1985 Revision): primary 20 M 05; secondary 20 M 07, 08 B 15.

Pseudovarieties of completely regular semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. If C is an isomorphism class of semigroups and ϱϵCon S, then we say that ϱ is over C if the idempotent ϱ-class belong to C. On Con S we can introduce the relations U, V, Tl, Tr and T as follows: if ϱ, θ, ϵConS, then we say that ϱ and θ are U− [V−, Tl−, Tr−, T−] related if both ϱ/ϱ∩θ and θ/ϱ∩gq over completely simple semigroups [rectangular band, left groups, right groups, groups]. It is shown that U, V, Tl, Tr and T are complete congruences on Con S.Various other characterizations of these congruences on Con S are obtained. Some of the congruences are studied for completely regular semigroups, orthodox semigroups and bands of groups. Further, since for any regular semigroup S, V∩Tl∩Tr is the identity relation, we obtain a subdirect decomposition of Con S.

Rees matrix semigroups over inverse semigroups

Three methods for the construction of all complete congruences on the lattice Lv(V) of subvarieties of a variety V are introduced. It is shown that there exists an order preserving embedding of the lattice of complete congruences on the lattice Lp(P) of all subpseudovarieties of a given pseudovariety P into the direct product of the lattices of complete congruences on lattices of subvarieties of varieties generated by members of P; thus there are methods for constructing all complete congruences on Lp(P). By way of application, 2ℵ0 complete congruences and complete endomorphisms are constructed on any lattice Lv(V), where V is a certain epigroup variety which includes all bands; there is an analogous application for the lattice of all pseudovarieties of semigroups.

POLYHEDRAL CONVEX CONES AND THE EQUATIONAL THEORY OF THE BICYCLIC SEMIGROUP

We shall show that several results concerning the lattice of completely regular semigroup varieties find their analogues for the lattice of pseudovarieties of completely regular semigroups. We establish several complete idempotent endomorphisms and a subdirect decomposition of this lattice of pseudovarieties. These investigations culminate in Theorem 18 which is the analogue for pseudovarieties of Polák’s description [19] of the lattice of completely regular semigroup varieties. We shall in particular be able to describe the lattice of pseudovarieties of orthogroups in terms of the lattice of pseudovarieties of groups.IfV is a variety and FinV the pseudovariety consisting of the finite members ofV, then we show that the lattice of pseudovarieties contained in FinV divides the ideal lattice of the lattice of varieties contained inV. This will entail in particular that the lattice of pseudovarieties of completely regular semigroups [monoids] is modular.

Finitely presented groups and completely regular semigroups

A Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

Completely regular semigroups in the variety generated by the bicyclic semigroup

To any given balanced semigroup identityv w a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup BC Dh a; b j a 2 b D aba D a; ab 2 D bab D bi

Semilattice transversals of regular bands I

Abstract It will be shown that every finitely presented group is the greatest group homomorphic image of a finitely presented completely regular semigroup which has a solvable word problem and whose maximal subgroups are either free froups or one-relator groups.