A concept of variety for regular biordered sets

We define a bivariety of regular biordered sets to be a nonempty class of regular biordered sets which is closed under taking direct products, regular bimorphic images and relatively regular biordered subsets. It is then shown that there is a complete lattice morphism mapping the complete lattice of all e-varieties of regular semigroups onto the complete lattice of all bivarieties of regular biordered sets; as a corollary we prove that there is a complete lattice morphism mapping the complete lattice of all e-varieties of E-solid regular semigroups onto the complete lattice of all varieties of solid binary algebras. Examples of bivarieties include the class of all solid regular biordered sets and the class of all local semilattices. For each setX with at least two elements, we show that a bivariety contains a free object onX if and only if it consists entirely of solid regular biordered sets or it consists entirely of local semilattices. The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award.     

The existence of free products in existence varieties of regular semigroups

It is shown that an existence varietyV of regular semigroups contains (all) free products if and only ifV consists solely of locally inverse orE-solid semigroups.Presented by B. M. Schein.The author is indebted to the Australian Research Council for financial support (ARC grant A69231516).     

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.     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.     

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 an assertion of J. Rhodes and the finite basis and finite vertex rank problems for pseudovarieties

J. Rhodes asserted at in Braga in 1997, in response to a question of J. Almeida, that is not finite vertex rank. We prove his assertion and more. By way of contrast, we show that is local, i.e. has vertex rank 1.     

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

Amalgamated products and properly 3-realizable groups

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π₁(K)≅G and whose universal cover has the proper homotopy type of a 3-manifold (with boundary).     

幺半群左正则纯整半直积上的同余

研究了幺半群半直积上的同余，给出了幺半群半直积的所谓同余分解定理，并特别讨论了幺半群左正则纯整半直积及其子类上的同余．     

Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP ₁ andP ₂, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP ₁ and whose vertex-figures are isomorphic toP ₂? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.     

McAlister's P-Theorem via Schützenberger Graphs

Abstract

We give a short proof, using Schützenberger graphs, of McAlister's P-theorem. The proof, when restricted to free inverse semigroups, turns into Munn's geometric multiplication of Munn trees.     

Centrality in Rees-Sushkevich varieties

Denote by RS _n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS _n are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C _m RS _n to be the class of all semigroups S in RS _n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup. The classes C _m RS _n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C ₂ RS _n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties C _m RS _n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C _m RS _n , where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C _∞ RS _n ) is non-finitely based. Further, the variety C _∞ RS _n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based. Received October 29, 2003; accepted in final form February 11, 2007.     

Realizations of regular apeirotopes

Summary In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.     

Scorza varieties and Jordan algebras

In this paper, I give two very direct proves of the correspondance between a geometric object (Scorza varieties) and an algebraic one (Jordan algebras). I also give a short proof of the homogeneity of Scorza varieties, and a new and very simple proof of properties of the automorphism group of a Jordan algebra.     

Graphs whose endomorphism monoids are regular

In this paper, we give several approaches to construct new End-regular (-orthodox) graphs by means of the join and the lexicographic product of two graphs with certain conditions. In particular, the join of two connected bipartite graphs with a regular (orthodox) endomorphism monoid is explicitly described.     

The Berman conjecture is true for nilpotent extensions of regular semigroups

No Abstract.  .     

Generalizing the Croke-Kleiner construction

It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also satisfy the stronger condition of being cell-like equivalent. This article describes a construction which will produce CAT(0) groups with multiple boundaries. These groups have very complicated boundaries in high dimensions. It is our hope that their study may provide insight into Bestvina's question.     

Classification and statistics of finite index subgroups in free products

The principal theme of this paper is the enumeration of finite index subgroups Δ in a free product Γ of finite groups under various restrictions on the isomorphism type of Δ. In particular, we completely resolve the realization, asymptotic, and distribution problems for free products Γ of cyclic groups of prime order (prior to this work, these questions were wide open even in the case of the classical modular group). This complex of problems, usually referred to as Poincaré-Klein Problem, originally arose around 1880 out of the work of Klein and Poincaré on automorphic functions and related number theory, but has also grown roots in geometric function theory and, more recently, in the theory of subgroup growth. Ideas and techniques from the theory of generalized permutation representations (an enumerative theory of wreath product representations recently developed by the first named author) play a fundamental role here. Other tools come from analytic number theory, combinatorics, and probability theory.