Narrow Results

This study aims to introduce the concept of $\alpha$-ideal in bounded commutative residuated lattices and establish some related properties. In this paper, we show that the set of $\alpha$-ideals of a bounded commutative residuated lattice is a Heyting algebra, and an algebraic lattice. Moreover, we state the prime $\alpha$-ideal theorem, and describe relations between $\alpha$-ideals and some types of ideals of a bounded commutative residuated lattice. Finally, we discuss correspondences between $\alpha$-ideals and $\alpha$-filters of a bounded commutative residuated lattice.

The concept of σ-ideals is introduced in almost distributive lattices (ADLs). Generalized stone ADLs are characterized in terms of their σ-ideals and α-ideals. Normal ADLs are also characterized in terms of their O-ideals and σ-ideals. Finally, a discussion is made about the epimorphic images and inverse images of σ-ideals.

The concepts of annulets and α-ideals are introduced in C-algebras. α-ideals are characterized in terms of annulets and minimal prime ideals. A set of equivalent conditions is established for every ideal of a C-algebra to become an α-ideal. Some topological properties of the class of all prime α-ideals are studied in a C-algebra.

In ordered semigroups, the notion of $\alpha$-ideals is a generalized concept of $(m,n)$-ideals, where $m,n\in {\mathbb{N}}_0$. We apply the notion of $\alpha$-ideals to characterize all regularities in ordered semigroups.

The concept of an Almost Distributive Lattice (ADL) has come into being in 1981. The development of the theory of Almost Distributive Lattices since its inception is given. The concepts of S– ideal in ADLs, normal ADLs, Topological representation of normal ADLs, conjunctively S–regular ADLs, S–normal ADLs, Heyting ADLs, and L–ADLs are studied.