Congruence Distributive Varieties Research Articles

Meet infinite distributivity for congruence lattices of direct sums of algebras

Abstract We study meet infinite distributivity (MID) in congruence lattices of direct sums of algebras. The main result of this note is that in a congruence distributive variety the MID of congruence lattices is preserved by direct sums of algebras, it means that the congruence lattice of a direct sum of algebras is MID if and only if a congruence lattice of every constituent algebra is MID.

Fraser–Horn–Hu property for ordered algebras

Fraser and Horn, and independently Hu, studied varieties of algebras satisfying the property that for every every congruence is a product congruence, i.e., for some i = 1, 2. Varieties of rings with identity and congruence distributive varieties of algebras satisfy this property. It is easy to show that an algebra has the Fraser–Horn–Hu property if and only if the map is a lattice isomorphism from to The property was later referred to in the literature as the Fraser–Horn–Hu property. It turns out that the Fraser–Horn–Hu property is a Mal’cev condition for varieties. In this paper, we generalize this property to varieties of ordered algebras. The classic result of Fraser, Horn and Hu follows as a special case.

Sheffer operation in relational systems

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system {mathbf {A}} with involution induces a Sheffer operation assigned to the twist product of {mathbf {A}}.

Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.

Diagram induced properties of congruence lattices

We investigate dual spaces of congruence lattices of algebras in a congruence-distributive variety $${\mathcal V}$$ . Our aim is to connect topological properties of these spaces with diagrams of finite $$(\vee ,0)$$ -semilattices liftable in $${\mathcal V}$$ . We achieve this aim for diagrams indexed by finite trees.

Existence of cube terms in finite algebras

We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most N, where the number N depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on N that, in the special case of algebras with more than \(\left( {\begin{array}{c}|A|\\ 2\end{array}}\right) \) basic operations, improves an earlier result of K. Kearnes and Á. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a k-ary near unanimity operation if and only if it contains a k-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.

The Tschantz and the alvin higher conditions are equivalent in congruence distributive varieties

We show that, under the assumption of congruence distributivity, a condition by S. Tschantz characterizing congruence modularity is equivalent to a variant of the classical Jónsson condition. Here equivalence is intended in a strong sense, to the effect that the corresponding sequences of terms have exactly the same length.

Another characterization of congruence distributive varieties

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

The Gumm level equals the alvin level in congruence distributive varieties

Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and J\'onsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without J\'onsson terms. We prove the quite unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by $1$. It follows that every $r$-modular congruence distributive variety is $r^2-r+2$-distributive.

The Jónsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $ J _{ \mathcal V} (m) $ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots $ holds in $\mathcal V$, with $m$ occurrences of $ \circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $ J _{ \mathcal V} (m) =k$, then $ J _{ \mathcal V} (m \ell ) \leq k \ell $, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $ J _{ \mathcal V} (1)=2 $, that is, $\mathcal V$ is $3$-distributive, then $ J _{ \mathcal V} (m) \leq m $, for every $m \geq 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $ J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are stated at various places.

A note on homomorphisms between products of algebras

Let {mathcal {K}} be a congruence distributive variety and call an algebra hereditarily directly irreducible (HDI) if every of its subalgebras is directly irreducible. It is shown that every homomorphism from a finite direct product of arbitrary algebras from {mathcal {K}} to an HDI algebra from {mathcal {K}} is essentially unary. Hence, every homomorphism from a finite direct product of algebras {mathbf {A}}_i (iin I) from {mathcal {K}} to an arbitrary direct product of HDI algebras {mathbf {C}}_j (jin J) from {mathcal {K}} can be expressed as a product of homomorphisms from {mathbf {A}}_{sigma (j)} to {mathbf {C}}_j for a certain mapping sigma from J to I. A homomorphism from an infinite direct product of elements of {mathcal {K}} to an HDI algebra will in general not be essentially unary, but will always factor through a suitable ultraproduct.

Uncountable critical points for congruence lattices

The critical point between two classes \({{\mathcal K}}\) and \({{\mathcal L}}\) of algebras is the cardinality of the smallest semilattice isomorphic to the semilattice of compact congruences of some algebra in \({{\mathcal K}}\), but not in \({{\mathcal L}}\). Our paper is devoted to the problem of determining the critical point between two finitely generated congruence-distributive varieties. For a homomorphism \({\varphi: S \rightarrow T}\) of \({(0, \vee)}\)-semilattices and an automorphism \({\tau}\) of T, we introduce the concept of a \({\tau}\)-symmetric lifting of \({\varphi}\). We use it to prove a criterion which ensures that the critical point between two finitely generated congruence-distributive varieties is less or equal to \({\aleph_{1}}\). We illustrate the criterion by constructing two new examples with the critical point exactly \({\aleph_{1}}\).

Naturally dualizable algebras omitting types 1 and 5 have a cube term

An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if A omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then A has a cube term.

Finitely Related Algebras In Congruence Distributive Varieties Have Near Unanimity Terms

Abstractwe show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. as a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. this consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

Congruence Distributivity Implies Bounded Width

We show that a constraint language with compatible Jonsson terms (or, equivalently, associated with an algebra generating a congruence distributive variety) defines a constraint satisfaction problem solvable by the local consistency checking algorithm.

CONGRUENCE LIFTING OF SEMILATTICE DIAGRAMS

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted Mn). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

The existence of a near-unanimity term in a finite algebra is decidable

AbstractWe prove that it is decidable of a finite algebra whether it has a near-unanimity term operation, which settles a ten-year-old problem. As a consequence, it is decidable of a finite algebra in a congruence distributive variety whether it admits a natural duality.

Algebraically closed algebras in certain small congruence distributive varieties

In a finitely generated congruence distributive variety satisfying a weak congruence extension property, the algebraically closed algebras are precisely updirected unions of maximal subdirectly irreducibles. The class of algebraically closed algebras of such a variety is elementary and definable by Horn sentences.

Polynomial functions on subdirect products

A congruence preserving function on a subdirect product of two finite Mal’cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A × B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A × B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A × B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.