The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.

A subobjects structure of the category $\Omega$- of $\Omega$-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\to )$ is investigated, where an $\Omega$-fuzzy set is a pair $𝐀=(A,\delta )$ such that $A$ is a set and $\delta A\times A\to \Omega$ is a special map. Special subobjects (called complete) of an $\Omega$-fuzzy set $𝐀$ which can be identified with some characteristic morphisms $𝐀\to {\Omega }^{*}=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms ${\neg }_{\Omega }{\Omega }^{*}\to {\Omega }^{*},{\cap }_{\Omega }$, ${\cup }_{\Omega }{\Omega }^{*}\times {\Omega }^{*}\to {\Omega }^{*}$ are characteristic morphisms of complete subobjects.

A meet semilattice with a partial join operation satisfying certain axioms is a JP-semilattice. A PJP-semilattice is a pseudocomplemented JP-semilattice. In this paper we describe the smallest PJP-congruence containing a kernel ideal as a class. Also we describe the largest PJP-congruence containing a filter as a class. Then we give several characterizations of congruence kernels and cokernels for distributive PJP-semilattices.

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...

Using congruence schemes we formulate new characterizations of congruence distributive, arithmetical and majority algebras. We prove new properties of the tolerance lattice and of the lattice of compatible reflexive relations of a majority algebra and generalize earlier results of H.-J. Bandelt, G. Cz'{e}dli and the present authors. Algebras whose congruence lattices satisfy certain 0-conditions are also studied.

It is known that congruence lattices of pseudocomplemented semilattices are pseudocomplemented [4]. Many interesting properties of congruences on pseudocomplemented semilattices were described by Sankappanavar in [4], [5], [6]. Except for other results he described congruence distributive pseudocomplemented semilattices [6] and he characterized pseudocomplemented semilattices whose congruence lattices are Stone, i.e. belong to the variety B₁ [5]. In this paper we give a partial solution to a more...