Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This generalizes the well-known given by Birkhoff lattices posets. We also study concepts of dual atoms, boolean elements, complemented elements and irreducible elements. prove that sets form semi-boolean algebras. show set lattice lattice.

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in re...

In this paper, we give several new lattice identities valid in nonmodular lattices such that a uniquely complemented lattice satisfying any of these identities is necessarily Boolean. Since some of these identities are consequences of modularity as well, these results generalize the classical result of Birkhoff and von Neumann that every uniquely complemented modular lattice is Boolean. In part...

We call a semigroup compactification T of a (discrete) semigroup S Boolean if the underlying space of T is zero-dimensional. The class of Boolean compactifications of S is a complete lattice, in a natural way. Motivated by Numakura’s theoerem, we prove that this lattice is isomorphic to the ideal lattice of the lattice of finite congruence relations of S. We describe in some detail the lattice ...

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size 2 א0 , the size of the lattice itself. We also prove that it is consistent that the lattice has chains of size 2 א0 , and in fact that these big chains occur in every in...

The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Miklös and Singhi. We study the fundamentals properties of such lattices and of a particular class of boolean functions defined on them.

We believe that the study of the notions of universal algebra modelled in an arbitarry topos rather than in the category of sets provides a deeper understanding of the real features of the algebraic notions. [2], [3], [4], [S], [6], [7], [13], [14] are some examples of this approach. The lattice Id(L) of ideals of a lattice L (in the category of sets) is an important ingredient of the categ...