The Essentially Equational Theory of Horn Classes

It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic (see [2]). In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known (see [1]). Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes are in fact (equivalent to) quasivarieties.     

Equational Classes of Totally Ordered Modal Lattices

A modal lattice is a bounded distributive lattice endowed with a unary operator which preserves the join-operation and the smallest element. In this paper we consider the variety CH of modal lattices that is generated by the totally ordered modal lattices and we characterize the lattice of subvarieties of CH. We also give an equational basis for each subvariety of CH.     

The Full Transformation Semigroups of Finite Rank are Amalgamation Bases for Finite Semigroups

In this article, we prove that the full transformation semigroup on a finite set is an amalgamation base for finite semigroups.     

序半群的正则同余类

本文的目的是给出一个序半群的子集能成为某个正则同余的同余类的刻面,同时我们可以容易看出[5]中的关于一般半群(没有序关系)的相应的结论仅是本文的结论的应用.     

On Finite Semigroups Embeddable in Inverse Semigroups

No Abstract. October 29, 1998     

Decidability of Equational Theories of Coverings of Semigroup Varieties

For every proper semigroup variety X, there exists a semigroup variety Y satisfying the following three conditions: (1) Y covers X, (2) if X is finitely based then so is Y, and (3) the equational theory of X is decidable if and only if so is the equational theory of Y. If X is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of X are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of X are locally finite and (2) X includes a subvariety Y whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.     

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup

Abstract

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.     

Finite Derivation Type for Semigroups and Congruences

We consider a congruence ρ on a semigroup S as a subsemigroup of the direct product S × S. We prove that if ρ has finite derivation type (FDT), then so does S.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.     

Finite Nilpotent Semigroups of Small Coclass

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this article a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.     

Comparing Classes of Finite Structures

We compare classes of structures using the notion of a computable embedding, which is a partial order on the classes of structures. Our attention is mainly, but not exclusively, focused on classes of finite structures. Also, a number of problems are formulated.     

Kaplansky Classes and Cotorsion Theories of Complexes

In this article we provide arguments for constructing Kaplansky classes in the category of complexes out of a Kaplansky class of modules. This leads to several complete cotorsion theories in such categories. Our method gives a unified proof for most of the known cotorsion theories in the category of complexes and can be applied to the category of quasi-coherent sheaves over a scheme as well as the category of the representations of a quiver.