E-varieties of regular semigroups,relatively bifree objects and fully invariant congruences

Analogous to the concept of a free object on a setX in a variety of algebras is the notion of a bifree object onX in an e-variety of regular semigroups. If an e-variety contains a bifree object onX, then a homomorphic image of that bifree object is itself bifree onX in some e-variety if and only if the corresponding congruence is fully invariant. Furthermore, the lattice of e-subvarieties of any locally inverse or E-solid e-variety ε is antiisomorphic with the lattice of all fully invariant congruences on the bifree object on a countably infinite setX in ε. We give a Birkhoff-type theorem for classes of locally inverse or E-solid semigroups, and we give an intrinsic test for whether or not a regular semigroup is bifree onX in the e-variety it generates.     

Weakly Strict Regular Semigroups

Weakly strict regular semigroups WS represent a generalization of strict regular semigroups. For the intersection of WS with some usual e-varieties of regular semigroups, such as left regular orthodox semigroups and similar e-varieties V, we provide WS ⋂ V with a basis of identities and forbidden semigroups. The latter, by exclusion, characterize the given e-variety. We do this also for WS. Coupled with some results of Hall, Churchill and Trotter, we characterize in this way also the e-varieties of completely regular semigroups as well as regular semigroups which are locally completely regular. The e-varieties studied are depicted in a diagram.     

The bifree regularE-solid semigroups

A model of the bifree regularE-solid semigroup on a non-empty setX is presented which is defined on the free ‘locally’ unary semigroupoid on the Cayley graph of the free group onX by means of the fully invariant congruence on the set of arrows which correspondes to the variety of completely regular semigroups. Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903. The result of the paper was announced at the Workshop on Semigroups, Formal Languages and Groups (NATO ASI No. 920872), York, 7–21 August, 1993     

Subgroups of free idempotent generated regular semigroups

In [6] it is shown that the maximal subgroups of the free idempotent generated regular semigroup which is determined by the biordered set of a completely O-simple semigroup are free. In this note we shall extend this result to a wider class of semigroups.     

On free spectra of finite completely regular semigroups and monoids

We show that a finite completely regular semigroup has a sub-log-exponential free spectrum if and only if it is locally orthodox and has nilpotent subgroups. As a corollary, it follows that the Seif Conjecture holds true for completely regular monoids. In the process, we derive solutions of word problems of free objects in a sequence of varieties of locally orthodox completely regular semigroups from solutions of word problems in relatively free bands.     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.     

Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q.     

Actions and partial actions of inductive constellations

Constellations were recently introduced by the authors as one-sided analogues of categories: a constellation is equipped with a partial multiplication for which ‘domains’ are defined but, in general, ‘ranges’ are not. Left restriction semigroups are the algebraic objects modelling semigroups of partial mappings, equipped with local identities in the domains of the mappings. Inductive constellations correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids.     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.     

Monoid varieties defined byx n+1 =x are local

We offer a new proof of the theorem in the title. In fact, we prove that for any varietyH of groups of finite exponent, the varietyCR(H) of all completely regular monoids with subgroups fromH, is local. The analogous result holds for pseudovarieties. A previously published proof of the theorem in the title has been found deficient.     

Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q. (Received 27 April 1999; in revised form 20 October 1999)     

On Bruck-Reilly extensions

We present a generalization for the procedure of taking Bruck-Reilly extensions, and we characterize abstractly the regular semigroups which can be obtained in this way. We shall in particular characterize the regular semigroups which can be obtained by considering the usual Bruck-Reilly extensions. Our procedure generalizes Munn’s construction [3] which in its turn combines ideas used by Bruck [1] and Reilly [4].     

Classification of some <Emphasis Type="Italic">τ</Emphasis>-congruence-free completely regular semigroups

Let τ be an equivalence relation on a semigroup. We introduce τ-congruence-free semigroups, extending the notion of congruence-free semigroups, and classify all completely regular semigroups which are τ-congruence-free, where τ is one of Green’s relations H,L\mathcal{H},\mathcal{L} and D\mathcal{D} respectively. Taking τ as H\mathcal{H} as well as D\mathcal{D}, this settles two open problems posed by M. Petrich and N.R. Reilly.     

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the ‘∨-premorphisms’ part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (J. Algebra 141:422–462, 1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for (ordered) ∧-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.