### $*$-median

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Characterizations for a pseudocomplemented modular join-semilattice with 0 and 1 and its ideal lattice to be a Stone lattice are given.

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a...

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 $\mathbf{A}=(A,\delta )$ such that $A$ is a set and $\delta \phantom{\rule{0.222222em}{0ex}}A\times A\to \Omega $ is a special map. Special subobjects (called complete) of an $\Omega $-fuzzy set $\mathbf{A}$ which can be identified with some characteristic morphisms $\mathbf{A}\to {\Omega}^{*}=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms ${\neg}_{\Omega}\phantom{\rule{0.222222em}{0ex}}{\Omega}^{*}\to {\Omega}^{*},{\cap}_{\Omega}$, ${\cup}_{\Omega}\phantom{\rule{0.222222em}{0ex}}{\Omega}^{*}\times {\Omega}^{*}\to {\Omega}^{*}$ 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 $\mathcal{V}$ of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every $A\in \mathcal{V}$ 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 $\mathcal{V}$ and embeddings between them. We believe that the strategy used here can...