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Research and Exposition in Mathematics -- Volume 32

   Enlarged Picture

Ivan Chajda, Helmut Lšnger

Directoids. An Algebraic Approach to Ordered Sets

viii+176 pages, soft cover, ISBN 978-3-88538-232-4, EUR 28.00, 2011

Order is a theoretical model of preference which is used in everyday life and has applications in economical, sociological, technical and natural sciences, and in particular in mathematics. The formal theory dealing with ordered sets as a mathematical concept was treated by a number of authors in papers and several monographs.

There are ordered sets with particular properties that can be considered as algebras. The best known examples are semilattices and lattices. An important class of ordered sets which generalize semilattices is the class of up- (or down-) directed sets, i.e. ordered sets in which every pair of elements has a common upper (or lower) bound. Directoids, the main mathematical concept studied in this monograph, are an algebraic version of up- (or down-) directed sets. A common upper (or lower) bound is assigned to every pair x, y of elements in such a way that it coincides with max(x,y) (or min(x,y)) in case x, y are comparable.

Hence, directoids are algebras with one binary operation, which is not necessarily associative or commutative. However, the corresponding operation can be characterized by several simple identities and hence the class of directoids forms a variety.

Directoids can be enriched by complementation, pseudocomplementation or relative pseudocomplementation. Such algebras serve as an algebraic axiomatization of certain non-classical logics, in particular the logic of quantum mechanics. The basic properties of directoids, their variety and several applications are studied in this monograph.


1 Preliminaries 7
2 The concept of a directoid 17
3 Varieties of directoids 27
4 λ-lattices 43
5 Pseudocomplemented directoids 49
6 Relatively pseudocomplemented directoids 57
7 Bounded directoids with an antitone involution 73
8 Directoids with sectional involutions 89
9 A non-associative generalization of MV-algebras 97
10 Weak MV-algebras 119
11 A representation of effect algebras by means of commutative directoids 141
12 Relational systems 159