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ABSTRACT
The theory of ordered sets is a sub-branch of the set theory that deals with the concept of order using the binary relations. The notions of order are present everywhere in mathematics and in many other scientific disciplines, as well as in the fields of engineering. The first part of this article covers the different types of order relations leading to the ordered spaces, on their particular elements and special subsets, the lattices (complete, bounded, distributive...) and applications between ordered spaces. The second part focuses on collections of subsets of a given ambient set by presenting the main properties, then the most used categories of collections. The notions are illustrated by examples and counter-examples.
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Jean-Charles PINOLI: Professor - École Nationale Supérieure des Mines de Saint-Étienne, Saint-Étienne, France
INTRODUCTION
Ordered set theory is a sub-branch of set theory that deals with the concept of order using binary relations. Notions of order are present everywhere in mathematics and many other scientific disciplines, as well as in various fields of engineering. The first part of this article deals with the different types of order relations leading to ordered spaces, their remarkable elements and particular subsets, and applications between ordered spaces. The second part deals with collections of subsets of a given ambient set, presenting the main properties, followed by the most commonly used categories of collections. The concepts presented are illustrated with examples and counter-examples.
Preamble
Ordered Set Theory is a sub-branch of Set Theory
The definition of a partially ordered set was clearly formulated by F. Hausdorff (1914), although the axioms that appear in the definition of an order relation had previously been considered by G. Leibniz (circa 1690). A precise definition of a totally ordered set was published by G. Cantor (1895).
The first lattice structure appeared implicitly in the mid-nineteenth century in the form of Boolean algebras (G. Boole, 1847), followed by the use of lattices in the algebraic approach to number theory by R. Dedekind (1894, 1897).
G. Birkhoff deserves the greatest credit for the first consistent developments in lattice theory. Birkhoff (1933, 1940, 1948).
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KEYWORDS
set theory | order relation | lattices | stacks
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