Distributive Congruence Research Articles

Exact-m-majority terms

Abstract We say that an idempotent term t is an exact-m-majority term if t evaluates to a, whenever the element a occurs exactly m times in the arguments of t, and all the other arguments are equal. If m < n and some variety 𝓥 has an n-ary exact-m-majority term, then 𝓥 is congruence modular. For certain values of n and m, for example, n = 5 and m = 3, the existence of an n-ary exact-m-majority term neither implies congruence distributivity, nor congruence permutability.

Merging Intuitionistic and De Morgan Logics

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety DH of all DH-algebras is congruence distributive. The lattice of all subvarieties of DH is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed.

Mitschke’s theorem is sharp

Mitschke showed that a variety with an m-ary near-unanimity term has Jónsson terms $$t_0, \dots , t _{2m-4} $$ witnessing congruence distributivity. We show that Mitschke’s result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an m-ary near-unanimity term. Finally we present some simple observations about terms with just one “dissenter”, a generalization of a minority term.

S-полигоны над вполне упорядоченным моноидом с модулярной решеткой конгруэнций

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

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.

Conditions of modularity of the congruence lattice of an act over a rectangular band

We describe acts over rectangular bands that have modular, distributive or linearly ordered congruence lattice. It turns out that such acts have at most 11 elements, and their congruence lattice has at most 300 elements. Furthermore, certain facts are established about the structure of acts with modular congruence lattice over an arbitrary semigroup and about the structure of the congruence lattice of an act over a rectangular band. The work is based on the description of acts over a completely (0-)simple semigroup obtained by Avdeev and Kozhukhov in 2000 and on the characterization of disconnected acts with modular or distributive congruence lattice by Ptakhov and Stepanova in 2013.

Facets of congruence distributivity in Goursat categories

We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.

On the set of principal congruences in a distributive congruence lattice of an algebra

Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ represents the inclusion $Q\subseteq D$ by principal congruences if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for every subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be (A1)-representable. We prove that every (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is chain-representable in the sense of a recent paper of G. Gratzer. Combining the results of this paper with another paper by the present author, it follows that every (A1)-representable finite distributive lattice is fully representable even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.

Pseudo-BCH Semilattices

In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.

On the least distributive lattice congruence on a semiring with a semilattice additive reduct

We characterize the least distributive lattice congruence on the semirings S with semilattice additive reduct in different ways arising from the non-symmetry and non-transitivity of the binary relation \({\longrightarrow}\) defined by: for \({a, b{\in}S}\), \({a{\longrightarrow}b}\) if \({b^{n}{\in}\overline{SaS}}\) for some positive integer n. Non-symmetry of the transitive closure \({{\longrightarrow}^{\infty}}\) of \({\longrightarrow}\) gives us \({M(a)=\{x{\in}S {\mid} a{\longrightarrow}^{\infty} x\}}\) and \({N(a)=\{x{\in} S {\mid} x {\longrightarrow}^{\infty} a\}}\) which are the principal completely semiprime k-ideal and the principal filter of S generated by a respectively. Both the relations \({\mathcal{M}}\) and \({\mathcal{N}}\) induced by M(a) and N(a) respectively are the least distributive lattice congruence on S. Again non-transitivity of \({\longrightarrow}\) yields an expanding family \({\{ {\longrightarrow}^{n}\}}\) of binary relations which again associates subsets Mn(a) and Nn(a) for all \({a{\in}S}\), and induces equivalence relations \({\mathcal{M}_{n}}\) and \({\mathcal{N}_{n}}\). Also the n-th power of the symmetric opening \({{\longleftrightarrow}}\) of \({\longrightarrow}\) gives us \({\Sigma_{n}(a)=\{x {\in} S \mid a {\longleftrightarrow}^{n} x\}}\) which induces the equivalence relation \({\sigma_{n}}\). We have characterized the semirings which are distributive lattices of \({\mathcal{M}_{n}(\mathcal{N}_{n}, \sigma_{n})}\) -simple semirings.

Properties of relatively pseudocomplemented directoids

The concept of a relatively pseudocomplemented directoid was introduced recently by the first author. It was shown that the class of relatively pseudocomplemented directoids forms a variety whose axiom system contains seven identities. The aim of this paper is three-fold. First we show that these identities are not independent and their independent subset is presented. Second, we modify the adjointness property known for relatively pseudocomplemented semilattices in the way which is suitable for relatively pseudocomplemented directoids. Hence, they can also be considered as residuated structures in a rather modified version. We also get two important congruence properties, namely congruence distributivity and $3$-permutability valid in the variety $\mathcal {V}$ of relatively pseudocomplemented directoids. Then we show some basic results connected with subdirect irreducibility in $\mathcal {V}$. Finally, we show another way how to introduce pseudocomplementation on directoids via relative pseudocomplementation.

On congruence distributivity of ordered algebras with constants

We define the order-congruence distributivity at 0 and order- congruence n-distributivity at 0 of ordered algebras with a nullary op- eration 0. These notions are generalizations of congruence distributiv- ity and congruence n-distributivity. We prove that a class of ordered algebras with a nullary operation 0 closed under taking subalgebras and direct products is order-congruence distributive at 0 iff it is order- congruence n-distributive at 0. We also characterize such classes by a Mal'tsev condition.

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.

A Subalgebra Intersection Property for Congruence Distributive Varieties

Abstract. We prove that if a finite algebra Agenerates a congruence distributive variety, then the subalgebras of the powers of Asatisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.

Iterative separation in distributive congruence lattices

Abstract In [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties.

ON THE COMPLEXITY OF SOME MALTSEV CONDITIONS

This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of two-generated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity.

Few subpowers, congruence distributivity and near-unanimity terms

We prove that for any variety \(\mathcal{V}\), the existence of an edge-term (defined in [1]) and Jonsson terms is equivalent to the existence of a near-unanimity term. We also characterize the idempotent Maltsev conditions which are defined by a system of linear absorption equations and which imply congruence distributivity.

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary.

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width.

Distributive congruence lattices of congruence-permutable algebras

We prove that every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ 1 bound is optimal, as we find a distributive algebraic lattice D with ℵ 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E.T. Schmidt from 1969. Furthermore, D may be taken as the congruence lattice of the free bounded lattice on ℵ 2 generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued ‘distances’ used by B. Jónsson in his proof of Whitman's Embedding Theorem. In particular, the semilattice of compact elements of D is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive 〈 ∨ , 0 〉 -semilattice is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.