Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties

We introduce new sufficient conditions for a finite algebraU to possess a finite basis of identities. The conditions are that the variety generated byU possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatizable. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.     

Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties

The aim of this paper is to prove that every finitely generated, arithmetical variety of finite type, in which every subdirectly irreducible algebra has linearly ordered congruences has a decidable first order theory of its finite members. The proof is based on a representation of finite algebras from such varieties by some quotients of special subdirect products in which sets of indices are partially ordered into dual trees. Then the result of M. O. Rabin about decidability of the monadic second order theory of two successors is applied.Presented by Stanley Burris.     

Varieties with decidable finite algebras I: Linearity

The aim of this paper is to prove that every congruence distributive variety containing a finite subdirectly irreducible algebra whose congruences are not linearly ordered has an undecidable first order theory of its finite members. This fills a gap which kept us from the full characterization of the finitely generated, arithmetical varieties (of finite type) having a decidable first order theory of their finite members. Progress on finding this characterization was made in the papers [14] and [15].Presented by Stanley Burris.     

Weakly diagonal algebras and definable principal congruences

An algebra is called weakly diagonal if every subuniverse of its square contains the graph of an automorphism. We show that every variety generated by a finite algebra with no proper subalgebras has a weakly diagonal generator. The result is applied in several ways and, in particular, to show that every arithmetical affine complete variety of finite type has equationally definable principal congruences. This paper is dedicated to Walter Taylor. Received February 22, 2005; accepted in final form June 3, 2005. Work of the first author was supported by grant No. 5368 from The Estonian Science Foundation.     

A finite basis theorem for quasivarieties

We consider varieties with the property that the intersection of any pair of principal congruences is finitely generated, and, in fact, generated by pairs of terms constructed from the generators of the principal components in a uniform way. We say that varieties with this property haveequationally definable principal meets (EDPM). There are many examples of these varieties occurring in the literature, especially in connection with metalogical investigations. The main result of this paper is that every finite, subdirectly irreducible member of a variety with EDPM generates a finitely based quasivariety. This is proved in Section 2. In the first section we prove that every variety with EDPM is congruence-distributive.Presented by George Gratzer.     

Semi-Nelson Algebras

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.     

Definable principal congruences in congruence distributive varieties

We present an algorithm that, given a finite algebraA generating a congruence distributive (CD) variety, determines whether this variety has first order definable principal congruences (DPC). In fact, DPC turns out to be equivalent to the extendability of the principal congruences of certain subalgebras of the algebras in HS(A ³). To verify this algorithm, we investigate combinatorial properties of finite subdirect powers ofA. Our theorem has a relatively simple formulation for arithmetical algebras. As an application, we obtain McKenzie's result that there are no nondistributive lattice varieties with DPC.Presented by A. Pixley.Finally I wish to thank E. Fried, R. W. Quackenbush and P. Pröhle for many helpful conversations (some ideas of the paper came up by considering weakly associative lattices), and to A. F. Pixley for raising the problem mentioned in the first section, which was the starting point of this investigation.     

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.     

Affine complete varieties are congruence distributive

An algebra is affine complete iff its polynomial operations are the same as all the operations over its universe that are compatible with all its congruences. A variety is affine complete iff all its algebras are. We prove that every affine complete variety is congruence distributive, and give a useful characterization of all arithmetical, affine complete varieties of countable type. We show that affine complete varieties with finite residual bound have enough injectives. We also construct an example of an affine complete variety without finite residual bound.? We prove several results concerning residually finite varieties whose finite algebras are congruence distributive, while leaving open the question whether every such variety must be congruence distributive. Received February 28, 1997; accepted in final form December 9, 1997.     

Nondefinability of projectivity in lattice varieties

The relationship of projectivity between two quotients in a lattice is shown not to be first-order definable, in any nondistributive lattice variety. The proof depends on a special kind of subdirect power construction that shows the existence of arbitrarily long non-shortenable projectivities in such a variety. A similar result holds for weak projectivities. Even so, weak projectivities of bounded length do suffice to determine principal congruences in any variety generated by a finite lattice.     

Decision problem for orthomodular lattices

This paper answers a question of H. P. Sankappanavar who asked whether the theory of orthomodular lattices is recursively (finitely) inseparable (question 9 in [10]). A very similar question was raised by Stanley Burris at the Oberwolfach meeting on Universal Algebra, July 15–21, 1979, and was later included in G. Kalmbach’s monograph [6] as the problem 42. Actually Burris asked which varieties of orthomodular lattices are finitely decidable. Although we are not able to give a full answer to Burris’ question we have a contribution to the problem.   Note here that each finitely generated variety of orthomodular lattices is semisimple arithmetical and therefore directly representable. Consequently each such a variety is finitely decidable. (For a generalization of this, i.e. a characterization of finitely generated congruence modular varieties that are finitely decidable see [5].) In section 3, we give an example of finitely decidable variety of orthomodular lattices that is not finitely generated. Received June 28, 1995; accepted in final form June 27, 1996.     

On Finite Basis Property for Joins of Varieties of Associative Rings

If all finitely generated rings in a variety of associative rings satisfy the ascending chain condition on two-sided ideals, the variety is called locally weak noetherian. If there is an upper bound on nilpotency indices of nilpotent rings in a variety, the variety is called a finite index variety. We prove that the join of a finitely based locally weak noetherian variety and a variety of finite index is also finitely based and locally weak noetherian. One consequence of this result is that if an associative ring variety is connected by a finite path in the lattice of all associative ring varieties to a finitely based locally weak noetherian variety then such variety is also finitely based and locally weak noetherian.     

序半群的一类正则同余

设S是有向序半群,本文给出了S上的一类正则同余,称为强序同余的定义及性质.证明了S的强序同余是强正则同余,但反之不成立.同时证明了强序同余格SOC(S)是S的同余格C(S)关于通常集合的交和传递积的V-完备的分配子格.     

On the congruence extension property

It is shown that a quasivariety has the extension property for all relative congruences if it has the extension property for principal relative congruences. This generalizes the analogous result for varieties, due to A. Day. The proof differs from Day's in that it avoids the use of Zorn's Lemma. Received February 13, 1997; accepted in final form January 19, 1998.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implicative twist-structures

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.     

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.     

Congruences on *-Regular Semigroups

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup. This revised version was published online in June 2006 with corrections to the Cover Date.     

Irreducible orthomodular lattices which are simple

It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.Presented by B. Jónsson.Supported by a grant from NSERC.