We say that a variety $𝒱$ of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every $A\in 𝒱$ is closed under intersection. We investigate the congruence lattices of algebras in locally finite, congruence-distributive CIP varieties and obtain a complete characterization for several types of such varieties. It turns out that our description only depends on subdirectly irreducible algebras in $𝒱$ and embeddings between them. We believe that the strategy used here can...

We present a countable infinite chain of conditions which are essentially weaker then congruence modularity (with exception of first two). For varieties of algebras, the third of these conditions, the so called 4-submodularity, is equivalent to congruence modularity. This is not true for single algebras in general. These conditions are characterized by Maltsev type conditions.

We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.

For a class $K$ of structures and $A\in K$ let ${Con}^{*}\left(A\right)$ resp. ${Con}^K\left(A\right)$ denote the lattices of $*$-congruences resp. $K$-congruences of $A$, cf. Weaver [25]. Let ${Con}^{*}\left(K\right):=I\{{Con}^{*}\left(A\right):A\in K\}$ where $I$ is the operator of forming isomorphic copies, and ${Con}^r\left(K\right):=I\{{Con}^K\left(A\right):A\in K\}$. For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${Con}^{<}\left(A\right)$, and let ${Con}^{<}\left(K\right):=I\{{Con}^{<}\left(A\right):A\in K\}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^s$ and $P$, respectively. Let $\lambda$ be a lattice identity and let $\Sigma$ be a set of lattice identities. Let $\Sigma {\vDash }_c\lambda (r;Q^s,P)$ denote...

We present a formal scheme which whenever satisfied by relations of a given relational lattice $L$ containing only reflexive and transitive relations ensures distributivity of $L$.

We obtain the construction of free abelian extensions in a congurence-permutable variety V using the construction of a free abelian extension in a variety of algebras with one ternary Mal'cevoperation and a monoid of unary operations. We also use this construction to obtain a free solvable V-algebra.

We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties....

We study $\vee$-semilattices and lattices with the greatest element 1 where every interval [p,1] is a lattice with an antitone involution. We characterize these semilattices by means of an induced binary operation, the so called sectionally antitone involution. This characterization is done by means of identities, thus the classes of these semilattices or lattices form varieties. The congruence properties of these varieties are investigated.