The Word Problem in the Variety of Inverse Semigroups with Abelian Covers

The variety of inverse semigroups which possess E-unitary coversover Abelian groups coincides with the Mal'cev product of thevariety of semilattices and the variety of Abelian groups,andalso with the variety generated by semidirect products of semilatticesand Abelian groups. We show that this variety (and any varietyof inverse semigroups that contains this variety) has undecidableword problem.     

E-unitaryR-unipotent semigroups

By making use of McAlister’s P-theorem [4] O’Carroll proved in [5] that every E-unitary inverse semigroup can be embedded into a semidirect product of a semilattice by a group. Recently an alternative proof of this result was published by Wilkinson [10]. In this paper we generalize this theorem by proving that every E-unitaryR-unipotent semigroup S can be embedded into a semidirect product of a band B by a group where B belongs to the variety of bands generated by the band of idempotents of S.     

P-反演半群的P-次直积

本文研究了P-反演半群的P-次直积和E-囿P-反演盖.利用P-反演半群的P-次直积的结构,引入了P-反演半群的E-囿P-反演盖概念,并刻画了它们的结构.最后,讨论了P-反演半群到给定群上的囿P-次同态,推广了文献[9]中的一些结果到P-反演半群上.     

E-酉正则半群

本文定义了PO-六元组的概念,给出了E-酉正则半群的一种结构.     

An Application of Groupoids of Fractions to Inverse Semigroups

We describe an application of category theory to the theory of inverse semigroups: we prove the P-theorem for E-unitary inverse semigroups using groupoids of fractions of their associated division categories.     

On Locally Inverse Semigroups Having Weakly E-unitary Covers

Recently, Billhardt has characterized the locally inverse semigroups embeddable in Rees matrix semigroups over generalized inverse semigroups. We prove here that these semigroups are just the locally inverse semigroups having weakly E-unitary covers.     

Proper Covers for Left Ample Semigroups

A left ample semigroup is a semigroup with a unary operation ⁺ which has a (2,1)-algebra embedding into a symmetric inverse monoid I(X), the operation ⁺ on I(X) being defined by α⁺ = αα^-1. We consider some analogues for left ample semigroups of results on E-unitary covers of inverse semigroups due to McAlister and Reilly. The analogue of an E-unitary cover is a proper cover, and we discuss the construction of proper covers in terms of relational homomorphisms, and of dual prehomomorphisms. We observe that our construction gives an E-dense proper cover for an E-dense left ample semigroup. We also consider proper covers constructed from strict embeddings into factorisable left ample monoids. In contrast to the inverse case, not all proper covers arise in this way. However, in the E-dense case, we characterise those E-dense proper covers which can be constructed from such embeddings.     

A Representation for $F$-Regular Semigroups

A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence _S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/_S, where X belongs to the band variety, generated by the band of idempotents E_S of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.     

Sur un théorème de connexité de Mumford pour les espaces homogènes

We prove some consequences of an old unpublished connectivity result of Mumford. These mostly deal with the fundamental group of some (ramified) coverings of homogeneous spaces; for example, it is shown that the fundamental group of a complex projective irreducible normal variety which is a covering of a simple (in the sense of 2.2) homogeneous space V, of degree ≤dim (V), is isomorphic to π₁ (V).     

Shmel’kin embeddings for abstract and profinite groups

The Magnus embedding is well known: given a group A=F/R, where F is a free group, the group F/[R, R] can be represented as a subgroup of a semidirect product AT, where T is an additive group of a free Z A-module. Shmel’kin genralized this construction and found an embedding for F/V(R), where V(R) is the verbal subgroup of R corresponding to a variety V. Later, he treated F as a free product of arbitrary groups, and on condition that R is contained in a Cartesian subgroup of the product, pointed out an embedding for F/V(R). Here, we combine both these Shmel’kin embeddings and weaken the condition on R, by assuming that F is a free product of groups A_i (iεI) and a free group X, and that its normal subgroup R has trivial intersection with each factor A_i. Subject to these conditions, an embedding for F/V(R) is found; we cell it the generalized Shmel’kin embedding. For the case where V is an Abelian variety of groups, a criterion is specified determining whether elements of AT belong to an embedded group F/V(R). Similar results are proved also for profinite groups. Supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 598–612, September–October, 1999.     

Embeddings of symmetric varieties

We generalize to the case of a symmetric variety the construction of the enveloping semigroup of a semisimple algebraic group due to E. B. Vinberg, and we establish a connection with the wonderful completion of the associated adjoint symmetric variety due to C. De Concini and C. Procesi.The author gratefully acknowledges the financial support of the Fonds FCAR and thanks the referees and V. Ginzburg for their comments.     

Definable principal subcongruences

For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group variety V(S ₃ ) does not have DPSC. Received May 9 2000; accepted in final form April 26, 2001.     

Derived categories of Fano threefolds

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 116–128. In memory of V.A. Iskovskikh     

Lower-Modular Elements of the Lattice of Semigroup Varieties

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.     

Subquasivarieties of regularized varieties

This paper considers the lattice of subquasivarieties of a regular variety. In particular we show that if V is a strongly irregular variety that is minimal as a quasivariety, then the smallest quasivariety containing both V and SI (the variety of semilattices) is never equal to the regularization V of V.We use this result to describe the lattice of subquasivarieties of V in several special but quite common, cases and give a number of applications and examples.Presented by R. W. Quackenbush.Work on this paper was begun while the second author was visiting Iowa State University during the summer of 1994.     

Soluble Groups and Varieties of <Emphasis Type="Italic">l</Emphasis>-Groups

A sufficient condition is given under which factors of a system of normal convex subgroups of a linearly ordered (l.o.) group are Abelian. Also, a sufficient condition is specified subject to which factors of a system of normal convex subgroups of an l.o. group are contained in a group variety . In particular, for every soluble l.o. group G of solubility index n, n ⩾ 2, factors of a system of normal convex subgroups are soluble l.o. groups of solubility index at most n − 1. It is proved that the variety of all lattice-ordered groups, approximable by linearly ordered groups, does not coincide with a variety generated by all soluble l.o. groups. It is shown that if is any o-approximable variety of l-groups, and if every identity in the group signature is not identically true in , then contains free l.o. groups.Supported by FP “Universities of Russia” grant UR. 04. 01. 001.__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 355–367, May–June, 2005.     

Finite Equivalence Relations on Algebraic Varieties and Hidden Symmetries

This paper can be considered as a continuation of Miyanishi's paper which contains a theorem on existence of a quotient of an affine normal or a projective smooth variety by a finite equivalence relation such that every component of the relation projects onto the variety (we call such an equivalence relation a wide finite equivalence relation). Later papers of Kollar and Keel-Mori shed new light on the subject and can serve as a base for further studies. The results of the present paper are based on the fact that every wide finite equivalence relation on a normal variety V is determined by an action of a finite group on the normalization of V in some Galois extension of k(V). Hence, such an equivalence relation hides some symmetry of a (ramified) cover of V. One may find some analogy of the situation with the concept of a hidden symmetry considered in physics. An important part of the paper is examples described in Section 6 which show that the main result of the paper (Theorem 2.3) is valid neither in the seminormal case, nor under the additional assumptions that there exists a finite morphism whose fibers contain equivalence classes of a given finite relation. In the nonnormal case, identification of some points described by a finite wide equivalence relation may force identification of some other nonequivalent points. This seems to show that the class of normal varieties and wide equivalence relation is a proper frame for considering the general problems of quotients by finite equivalence relations.     

On orbit closures of spherical subgroups in flag varieties

Let be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on with finitely many orbits, and let V be an H-orbit closure in . Expanding the cohomology class of V in the basis of Schubert classes defines a union V₀ of Schubert varieties in with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V₀ is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V₀ in . Thus, for any effective line bundle L on , the restriction map is surjective, and for all . Received: April 17, 2000