We follow throughout a heterogeneous approach to present some characterizations (Theorem 3.5 and Theorem 3.35) of subdirectly irreducible bijective Boolean modules. The notions of essential element (as modular element) and [1]-closed element (as Boolean element) were implemented to this class of algebras. Our final aim of reaching the class of simple bijective Boolean modules, as a particular c...

The size of a concept lattice may increase exponentially with the size of the context. When the number of nodes is too large, it becomes very difficult to generate and study such a concept lattice. A way to avoid this problem is to break down the lattice into small parts. In the subdirect decomposition, the small parts are factor lattices which are meaningful in the Formal Concept Analysis (FCA...

In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particular five-element directoid. Roughly speaking, congruence rich varieties are those in which all la...

Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to logic, like the guarded fragment or dynamic logic. Tarskian algebraic logic is far too broad and too fruitf...

Let n ≥ 3. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an n-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an n-di...

In the present paper we will ask for the lattice L(MVEx) of subvarieties of the variety defined by the set Ex(MV) of all externally compatible identities valid in the variety MV of all MV-algebras. In particular, we will find all subdirectly irreducible algebras from the classes in the lattice L(MVEx) and give syntactical and semantical characterization of the class of algebras defined by P -co...

Let $f : A rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we give a necessary and sufficient condition for the amalgamated algebra along an ideal $Abowtie^fJ$ to be $J$-Noetherian. Then we give a characterization for pseudo-irreducible ideals of $Abowtie^fJ$, in special cases.

It is well known that there are five classes of sloops of cardinality 16 " SL(16)s" according to the number of sub-SL(8)s [4, 6]. In this article, we will show that there are exactly 8 classes of nonsimple sloops and 6 classes of simple sloops of cardinality 20 "SL(20)s". Based on the cardinality and the number of (normal) subsloops of SL(20), we will construct in section 3 all possible classes...