Finite bases for finitely generated,relatively congruence distributive quasivarieties

In [21], D. Pigozzi has proved in a non-constructive way that every relatively congruence distributive quasivariety of finite type generated by a finite set of finite algebras is finitely axiomatizable. In this paper we show that the non-constructive parts of Pigozzi's argument can be replaced by constructive ones. As a result we obtain a method of constructing a finite set of quasi-equational axioms for each relatively congruence distributive quasivariety generated by a given finite set of finite algebras of finite type. The method can also be applied to finitely generated congruence distributive varieties.Presented by Joel Berman.     

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.     

Bounding essential arities of term operations in finitely decidable varieties

Let A be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical \({\tau}\). We provide an improved bound on the number of variables in which a term can be sensitive to changes within \({\tau}\). We utilize a multisorted algebraic construction, amalgamating the methods developed by Valeriote and McKenzie for the investigation of strongly abelian locally finite decidable varieties with those of Idziak for locally finite congruence modular finitely decidable varieties.     

Amalgamation classes of some distributive varieties

This paper investigates the amalgamation classes of finitely generated varieties with distributive congruence lattices. Necessary and sufficient conditions are given for an algebra to be a member of the amalgamation class of a variety generated by a finite modular lattice or pseudocomplemented distributive lattice and of a filtral variety.Presented by S. Burris.     

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.     

Ultraproducts preserve finite subdirect reducibility

An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties.     

On congruence modularity in varieties of logic

An open problem in algebraic logic asks whether a variety that is the equivalent algebraic semantics of an algebraizable logic must be congruence modular. A negative solution is provided here: it is shown that such ‘varieties of logic’ need satisfy no special congruence lattice identity. Received June 15, 1999; accepted in final form July 10, 2000.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Projective algebras and primitive subquasivarieties in varieties with factor congruences

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.     

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.     

Varieties with decidable finite algebras II: Permutability

This paper is a continuation of [3]. Congruence permutability is shown to be a necessary condition for a locally finite congruence distributive variety to have a decidable first order theory of its finite algebras. This is a positive answer to Problem 6 of S. Burns and H. P. Sankappanavar [2]. Moreover this allows us to give a full characterization of finitely generated congruence distributive varieties of finite type with decidable first order theories of their finite members.Presented by Stanley Burris.     

Quasiorder lattices of varieties

The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.     

Algebraic coalitions

The idea of an algebra in is introduced. Within congruence modular varieties such algebras are shown to be the abelian algebras with a one-element subalgebra. This leads on to the notion of algebraic coalition, which is characterized for congruence modular varieties and for varieties of Jónsson–Tarski algebras. This characterization displays an intimate relationship between algebraic coalitions, Gumm difference terms, and the centre of an algebra. Received July 16, 1996; accepted in final form May 2, 1997.     

Finitary decidability implies congruence permutability for congruence modular varieties

We show that if a locally finite congruence modular varietyV is finitely decidable, thenV has to be congruence permutable.Presented by S. Burris.     

Properties of Tense Logics

Based on the results of [11] this paper delivers uniform algorithms for deciding whether a finitely axiomatizable tense logic

• has the finite model property,
• is complete with respect to Kripke semantics,
• is strongly complete with respect to Kripke semantics,
• is d-persistent,
• is r-persistent.

It is also proved that a tense logic is strongly complete iff the corresponding variety of bimodal algebras is complex, and that a tense logic is d-persistent iff it is complete and its Kripke frames form a first order definable class. From this we obtain many natural non-d-persistent tense logics whose corresponding varieties of bimodal algebras are complex. Mathematics Subject Classification: 03B45, 03B25.     

Absolute retracts and essential extensions in congruence modular varieties

This paper studies absolute retracts in congruence modular varieties of universal algebras. It is shown that every absolute retract with finite dimensional congruence lattice is a product of subdirectly irreducible algebras. Further, every absolute retract in a residually small variety is the product of an abelian algebra and a centerless algebra.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

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.     

Separation in distributive congruence lattices

We define separable sets in algebraic lattices. For a finitely generated congruence distributive variety V \mathcal{V} , we show a close connection between non-separable sets in congruence lattices of algebras in V \mathcal{V} and the structure of subdirectly irreducible algebras in V \mathcal{V} . We apply the general results to some lattice varieties.