# Pseudo-complements

• Sambasiva Rao Mukkamala
Chapter

## Abstract

In mathematics, particularly in order theory, a pseudo-complement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element $$x\in L$$ is said to have a pseudo-complement if there exists a greatest element $$x^{*}\in L$$, disjoint from x, with the property that $$x\wedge x^{*} = 0$$. More formally, $$x^{*} = \max \{y\in L~|~x\wedge y = 0\}$$. The lattice L itself is called a pseudo-complemented lattice if every element of L is pseudo-complemented. Every pseudo-complemented lattice is necessarily bounded; i.e., it has a 1 as well. Since the pseudo-complement is unique by definition (if it exists), a pseudo-complemented lattice can be endowed with a unary operation * mapping every element to its pseudo-complement. The theory of pseudo-complements in lattices, and particularly in distributive lattices, was developed by M.H. Stone (Trans Am Math Soc 40:37–111, 1936), [228], O. Frink (Duke Math J 29:505–514, 1962), [97], and G. Gratzer (General Lattice Theory, Academic Press, New York 1978), [103]. Later many authors like R. Balbes (Distributive Lattices, University of Missouri Press, Columbia, 1974), [12], O. Frink (Duke Math J 29:505–514, 1962), [97] extended the study of pseudo-complements to characterize Stone lattices. In 2013, Cilo$$\breve{g}$$lu and Ceven (Algebra, 1–5, 2013), [53] studied the properties of the elements $$x*0$$ in a commutative and bounded BE-algebras. Recently in 2014, R. Borzooei et. al (J Math Appl 37:13–26, 2014), [23] studied some structural properties of bounded and involutory BE-algebras and investigate the relationship between them.