Congruence Distributive Varieties Research Articles

Optimal natural dualities. II: General theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R R of finitary algebraic relations yields a duality on a class of algebras A = I S P ⁡ ( M _ ) \mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M}) , those subsets R ′ R’ of R R which yield optimal dualities are characterised. Further, the manner in which the relations in R R are constructed from those in R ′ R’ is revealed in the important special case that M _ \underline {M} generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on M _ \underline {M} . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties B n \mathbf {B}_{n} of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.

The syntax and semantics of entailment in duality theory

AbstractBoth syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

The quest for strong dualities

AbstractWe give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

Skeletons of congruence distributive varieties of algebras

Skeletons of congruence distributive varieties of algebras

The determination of subvarieties of certain congruence-distributive varieties

This note concerns the problem of determining equational bases for the subvarieties of a congruence-distributive variety generated by a finite algebra. It describes a procedure, based on the duality for finite distributive lattices, which converts a local solution to this problem into a global one. For many varieties of distributive-lattice-ordered algebras a local solution of the required form can be obtained from a natural duality in which identities are very conveniently encoded.

Finite algebras that generate an injectively complete modular variety

We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

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.

Congruence distributive varieties of rings

Congruence distributive varieties of rings

Varieties with cofinal sets: examples and amalgamation

A variety $\mathcal {V}$ has a cofinal set $S \subset \mathcal {V}$ if any $A \in \mathcal {V}$ is embeddable in a reduced product of members of $S$. Amalgamation in and examples of such varieties are considered. Among other results, the following are proved: (i) every lattice is embeddable in an ultraproduct of finite partition lattices; (ii) if $\mathcal {V}$ is a residually small, congruence distributive variety whose members all have one-element subalgebras, then the amalgamation class of $\mathcal {V}$ is closed under finite products.

Invariant subspaces of some functional spaces on the group of motions of the Euclidean plane

i. A. G. Pinus, "On the relations of epimorphism and embeddability on congruence-distributive varieties," Algebra Logika, 24, NO. 5, 588-607 (1985). 2. A. G. Pinus, Congruence-Modular Varieties of Algebras [in Russianl, Irkutsk State Univ. (1986). 3. A. G. Pinus, "On simple epimorphism skeletons of congruence-distributive varieties," Izv. Vyssh. Uchebn. Zaved., Mat., No. ii, 67-70 (1987). 4. A. G. Pinus, "On coverings in epimorphism skeletons of varieties of algebras," Algebra Logika, 27, No. 3, 316-326 (1988). 5. A. G. Pinus, "ConGruence-distributive varieties of algebras," Progress in Science. Algebra, Geometry, and Toplogy [in Russian], Vol. 26, Bsesoyuz. Inst. Nauch. i Tekh. Informatsii, Moscow (1988), pp. 45-83. 6. A. G. Pinus, "On the number of incomparables in countable epimorphism skeletons of discriminator varieties," Algebra Logika, 2_88, No. 3, 311-323 (1989). 7. G. Gritzer, Universal Algebra, 2nd Ed., Springer-Verlag, Berlin-Heidelberg (1979). 8. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York-Heidelberg-Berlin (1981). H. Werner, Discriminator Algebras, Akademie Verlag, Berlin (1978). G. A. Fraser and A. Horn, "Congruence relations in direct product," Proc. Am. Math. Soc., 26, No. 3, 390-394 (1970). .

Congruence-distributive varieties of algebras

A survey of results on congruence-distributive varieties of algebras, concentrated on two directions: 1) the construction of algebras of congruence-distributive varieties and the structure of the variety itself, 2) questions connected with formalized (equational, elementary, etc.) theories of such varieties. In addition the survey includes papers connected with congruence-distributive varieties of algebras reviewed in RZh Matematika during the period 1976–1985.

On intervals and chains in epimorphness skeletons of congruence-distributive varieties

On intervals and chains in epimorphness skeletons of congruence-distributive varieties

Minimal varieties and quasivarieties

AbstractWe prove that every locally finite, congruence modular, minimal variety is minimal as a quasivariety. We also construct all finite, strictly simple algebras generating a congruence distributive variety, such that the sett of unary term perations forms a group. Lastly, these results are applied to a problem in algebraic logic to give a sufficient condition for a deductive system to be structurally complete.

Absolute Retracts and Amalgamation in Certain Congruence Distributive Varieties

AbstractIt is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.

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.

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].

On the Relationship of AP, RS and CEP in Congruence Modular Varieties. II

Let $V$ be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If $V$ is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented.

On the relationship of AP, RS and CEP in congruence modular varieties. II

Let V V be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If V V is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented.

Finite equational bases for congruence modular varieties

In this paper it is proved that a variety generated by a finite algebraic system with finitely many operations is finitely axiomatizable, provided that the variety is congruence modular and residually small. This result is an extension to congruence modular varieties of a well known theorem for congruence distributive varieties, due to K. A. Baker. Also, under somewhat less restrictive hypotheses, (which are satisfied by finite groups and rings) it is proved that a finite algebraic system belongs to a finitely axiomatizable locally finite variety.

On the relationship of AP, RS and CEP in congruence modular varieties

A condition is found on a congruence modular variety, guaranteeing that the implication AP & RS⇒CEP holds. The condition is in terms of the commutator on congruence lattices. In particular, the implication holds for any congruence distributive variety whose free algebra on four generators is finite.