Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.Presented by Joel Berman.To the memory of Basia Czelakowska.     

Subdirectly Irreducible Double Demi-p-Lattices

The aim of this paper is to give a characterization of the (finitely) subdirectly irreducible double demi-p-lattices. First, we prove a congruence representation theorem for double demi-p-lattices, which is a natural analogue of the theorem given in [2] for double p-algebras. These results are inspired by the representation theorem given by Lakser [6] for p-algebras, and yield a natural approach to the study of subdirectly irreducible algebras.     

Congruence Permutable Symmetric Extended de Morgan Algebras

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, Kn,0-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras.     

K11-代数的主同余关系的可补性

研究K11-代数的主同余关系的可补性,给出K11-代数的主同余关系是可补的充要条件。     

Interpolation properties in the extensions of the logic of inequality

We consider the modal logics wK4 and DL as well as the corresponding weakly transitive modal algebras and DL-algebras. We prove that there exist precisely 16 amalgamable varieties of DL-algebras. We find a criterion for the weak amalgamation property of varieties of weakly transitive modal algebras, solve the deductive interpolation problem for extensions of the logic of inequality DL, and obtain a weak interpolation criterion over wK4.     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Subreducts of MV-algebras with product and product residuation

Recently, MV-algebras with product have been investigated from different points of view. In particular, in [EGM01], a variety resulting from the combination of MV-algebras and product algebras (see [H98]) has been introduced. The elements of this variety are called ŁΠ-algebras. In this paper we treat subreducts of ŁΠ-algebras, with emphasis on quasivarieties of subreducts whose basic operations are continuous in the order topology. We give axiomatizations of the most interesting classes of subreducts, and we connect them with other algebraic classes of algebras, like f-rings and Wajsberg hoops, as well as to structures of co-infinitesimals of ŁΠ-algebras. In some cases, connections are given by means of equivalences of categories.Dedicated to the Memory of Wim BlokReceived June 19, 2002; accepted in final form November 29, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Semivarieties of nilpotent groups

Semivarieties of groups are quasivarieties defined by quasi-identities of the form t = 1 → f = 1. It is proved that a set of semivarieties in every variety of class two nilpotent p-groups of finite exponent having a commutator subgroup of exponent p (p is a prime) is at most countable. It is stated that a variety of class two nilpotent groups with commutator subgroup of exponent p contains a set of semivarieties of the cardinality of the continuum.     

Principal congruences of double demi-p-lattices

A formula is given to express a principal congruence on a double demi-p-lattice as a join of countably many principal lattice congruences. It is then applied to show that the variety of double demi-p-lattices has the congruence extension property. As special cases one obtains some known results for distributive doublep-lattices due to T. Hecht and T. Katriák.Presented by Bjarni Jónsson.     

Property (T) for topological groups and <Emphasis Type="Italic">C</Emphasis>*-algebras

The aim of the present (mostly expository) paper is to show the relationship of a generalization of Kazhdan’s property (T) for C*-algebras introduced in our recent paper to that of B. Bekka. It is shown that our definition coincides with Bekka’s definition for group C*-algebras of locally compact groups, whereas, in general, these definitions are distinct. Criteria for a C*-algebra to possess our property (T) are given. A number of examples of C*-algebras with and without property (T) are considered. Relations to K-theory are studied. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 171–192, 2007.     

Perturbations of C*-Algebraic Invariants

Kadison and Kastler introduced a metric on the set of all C*-algebras on a fixed Hilbert space. In this paper structural properties of C*-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison’s similarity problem transfers to close C*-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine K-theory and traces of close C*-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.     

Equimorphy in varieties of distributive double p-algebras

Any finitely generated regular variety V of distributive double p-algebras is finitely determined, meaning that for some finite cardinal n(V), any subclass S V of algebras with isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p-algebras must be almost regular.     

Approximately central matrix units and the structure of noncommutative tori

The property of approximate divisibility forC*-algebras is introduced and studied. Simple approximately divisibleC*-algebras are shown to have nice nonstableK-theory properties. Non-rational noncommutative tori are shown to be approximately divisible. It follows that every simple noncommutative torus (in particular, every irrational rotation algebra) has stable rank one and real rank zero.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Relative principal congruences in congruence-modular quasivarieties

The problem of definability of relative principal congruences in relatively congruence modular (RCM) quasivarieties is investigated. The RCM quasivarieties are characterized in terms of parameterized families of finite sets of pairs of terms which define relative principal congruences. Received April 24, 1995; accepted in final form March 2, 1998.     

Characterizable classes of lattices

We look at the concept of a characterizable class of systems. It is proved that there exist characterizable lattice varieties whose join in the lattice of all lattice varieties is not a characterizable variety. We point out two finitely characterizable lattice quasivarieties, which are not varieties, whose meet in the lattice of lattice quasivarieties is a variety. Also an example of a characterizable locally finite lattice variety is constructed.     

Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II

A general characterization of lattices of varieties and lattices of quasivarieties in terms of (quasi)Birkhoff classes is given and a method for constructing their homomorphic images is presented. As an application, it is proved that the lattice of varieties of modular lattices has a complete homomorphism onto the Boolean lattice of subsets of a countable set. Also, sufficient conditions are found for embedding the free lattice with generators in a given lattice of quasivarieties, and we show that these are also sufficient for a quasivariety to be Q-universal. Other applications and examples are given.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 369–397, July-August, 1995.