We investigate conditions for the existence of relative complements in ordered sets. For relatively complemented ordered sets with 0 we show that each element b ≠ 0 is the least one of the set of all upper bounds of all atoms contained in b.
The concept of relative pseudocomplement is introduced in a commutative directoid. It is shown that the operation of relative pseudocomplementation can be characterized by identities and hence the class of these algebras forms a variety. This variety is congruence weakly regular and congruence distributive. A description of congruences via their kernels is presented and the kernels are characterized as the so-called -ideals.
We characterise those Hilbert algebras that are relatively pseudocomplemented posets.
We extend the notion of a relatively pseudocomplemented meet-semilattice to arbitrary posets. We show some properties of the binary operation of relative pseudocomplementation and provide some corresponding characterizations. We show that relatively pseudocomplemented posets satisfying a certain simple identity in two variables are join-semilattices. Finally, we show that every relatively pseudocomplemented poset is distributive and that the converse holds for posets satisfying the ascending chain...
We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...
In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.