It is well known that every complete lattice can be considered as a complete lattice of closed sets with respect to appropriate closure operator. The theory of q-lattices as a natural generalization of lattices gives rise to a question whether a similar statement is true in the case of q-lattices. In the paper the so-called M-operators are introduced and it is shown that complete q-lattices are q-lattices of closed sets with respect to M-operators.

In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of $R$-polars are studied. Connections between $R$-polars and prime ideals, especially in distributive sets, are found.

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $\mathcal{A}$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $\mathcal{D}\left(A\right)$ of all deductive systems on $\mathcal{A}$. Moreover, relative annihilators of $C\in \mathcal{D}\left(A\right)$ with respect to $B\in \mathcal{D}\left(A\right)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $\mathcal{D}\left(A\right)$.

The aim of this paper is to transfer the concept of pseudocomplement from lattices to ordered sets and to prove some basic results holding for pseudocomplemented ordered sets.

Distributive ordered sets are characterized by so called generalized annihilators.

In [1] ideals and congruences on semiloops were investigated. The aim of this paper is to generalize results obtained for semiloops to the case of left divisible involutory groupoids.

A distributive pseudocomplemented set $S$ [2] is called Stone if for all $a\in S$ the condition $LU({a}^{*},{a}^{**})=S$ holds. It is shown that in a finite case $S$ is Stone iff the join of all distinct minimal prime ideals of $S$ is equal to $S$.

The paper shows that commutative Hilbert algebras introduced by Y. B. Jun are just J. C. Abbot’s implication algebras.

We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels.

We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.

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 characterize totally ordered sets within the class of all ordered sets containing at least four-element chains. We use a simple relationship between their isotone transformations and the so called 1-endomorphism which is introduced in the paper. Later we describe 1-, 2-, 3-, 4-homomorphisms of ordered sets in the language of super strong mappings.

Download Results (CSV)