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.     

Completions of orthomodular lattices

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.     

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.     

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)     

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.     

Adelic version of Margulis arithmeticity theorem

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one and set . Let S be any subset of R. For each , let be a connected semisimple adjoint -group and be a compact open subgroup for each finite prime . Let denote the restricted topological product of 's, with respect to 's. Note that if S is finite, . We show that if , any irreducible lattice in is a rational lattice. We also present a criterion on the collections and for to admit an irreducible lattice. In addition, we describe discrete subgroups of generated by lattices in a pair of opposite horospherical subgroups. Received: 30 November 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

On a subvariety of semi-De Morgan algebras

A characterization of principal congruences of the subvariety C of semi-DeMorgan algebras is given. This characterization is applied to determine the subdirectly irreducible algebras of the variety C and to describe a poset such that the lattice of its order ideals is isomorphic to the lattice of subvarieties of C. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

A characterization of state spaces of orthomodular lattices

It is shown that every rational polytope is affinely equivalent to the set of all states of a finite orthomodular lattice, and that every compact convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states of an orthomodular lattice.     

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.     

Congruences on near-Heyting algebras

A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

THE LATTICE OF CONGRUENCES OF A SUBDIRECTLY IRREDUCIBLE EXTENDED OCKHAM ALGEBRA

ABSTRACT

Here we investigate the structure of the lattice of congruences of a subdirectly irreducible extended Ockham algebra. In particular, we determine precisely when it is a chain.     

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.     

并素元有限生成格的弱直积分解

本文建立了并素元有限生成格的弱直积分解,并给出一个解决并素元生成的完全Heyting代数的直积分解问题的新方法;作为弱直积分解的应用,证明了并素元有限生成的完全Heyting代数必然同构于有限个既约的完全Heyting代数的直积,证明了并素元有限生成格是Boole代数的充要条件是它同构于某有限集的幂集格.     

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.     

Congruence computations in principal arithmetical varieties

This paper is a continuation of the earlier paper by the same authors in which a primary result was that every arithmetical affine complete variety of finite type is a principal arithmetical variety with respect to an appropriately chosen Pixley term. The paper begins by presenting an extension of this result to all finitely generated congruences and, as an example, constructs a closed form solution formula for any finitely presented system of pairwise compatible congruences (the Chinese remainder theorem). It is also shown that in all such varieties the meet of principal congruences is also principal, and finally, if a minimal generating algebra of the variety is regular, it is shown that the variety is also regular and the join of principal congruences is again principal.     

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.