Complemented lattice

A complemented lattice is a type of lattice that is both bounded and has a complement for every element. This means that for any element x in the lattice, there exists another element y such that their join (least upper bound) is the lattice's maximum element and their meet (greatest lower bound) is the lattice's minimum element. In other words, every element in a complemented lattice has a unique "opposite" element that, when combined, result in the lattice's maximum and minimum elements.From: Clonal sets of a binary relation [2018]

Stochastic Processes: An Introduction

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Published in Andrey Popoff, Fundamentals of Signal Processing in Metric Spaces with Lattice Properties, 2017

Andrey Popoff

Lattice L $ \mathbf L $ with zero O $ \mathbf O $ and unity I $ \mathbf I $ is called complemented lattice, if every element has a relative complement in the interval [O,I] $ [\mathbf O ,\mathbf I ] $ . Relative complements in the interval [O,I] $ [\mathbf O ,\mathbf I ] $ are simply called complements.

Clonal sets of a binary relation

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Published in International Journal of General Systems, 2018

Lemnaouar Zedam, Raúl Pérez-Fernández, Hassane Bouremel, Bernard De Baets

A (non-empty) poset is called a lattice if any two elements x and y have a greatest lower bound, denoted by and called the infimum of x and y, and a smallest upper bound, denoted by and called the supremum of x and y. Similarly, a lattice is called complete if every subset A of L has both a greatest lower bound, denoted by and called the infimum of A, and a smallest upper bound, denoted by and called the supremum of A. A lattice is called bounded if it has a smallest and a greatest element, respectively denoted by 0 and 1. A non-empty subset M of a lattice is called a sublattice of L if, for any , it holds that and . A complemented lattice is a bounded lattice in which any element x has a complement, i.e. there exists an element such that and . The relevant notions of distributivity and modularity of a lattice are characterized by means of the lattices and illustrated in Figure 5. In particular, a lattice is modular if and only if it has no sublattice of the form , and distributive if and only if it has no sublattice of the form or . For more details, we refer to Davey and Priestley (2002).

Complemented lattice

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