Pseudo-complements

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.

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsMVGR College of EngineeringVizianagaramIndia

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