Research and Exposition in Mathematics -- Volume 32
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.