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Introduction to Lattices and Order is a mathematical textbook on order theory by Brian A. Davey and Hilary Priestley. It was published by the Cambridge University Press in their Cambridge Mathematical Textbooks series in 1990, [ 1 ] [ 2 ] [ 3 ] with a second edition in 2002.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
The converse of this implication leads to functions that are order-reflecting, i.e. functions f as above for which f(a) ≤ f(b) implies a ≤ b. On the other hand, a function may also be order-reversing or antitone, if a ≤ b implies f(a) ≥ f(b). An order-embedding is a function f between orders that is both order-preserving and order ...
Distributivity is a basic concept that is treated in any textbook on lattice and order theory. See the literature given for the articles on order theory and lattice theory. More specific literature includes: G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical Society, 3: 677 - 680, 1952.
The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function X → {1}. Going further, even complete lattices can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.
Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. Taylor, Paul (1999), Practical foundations of mathematics, Cambridge Studies in Advanced Mathematics, vol. 59, Cambridge University Press, Cambridge, ISBN 0-521-63107-6, MR 1694820; Frenchman, Zack; Hart, James (2020), An Introduction to Order Theory, AMS
Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. Carl Hewitt; Henry Baker (August 1977). "Actors and Continuous Functionals" (PDF). Proceedings of IFIP Working Conference on Formal Description of Programming Concepts. Archived (PDF) from the original on April 12, 2019.
The Dedekind–MacNeille completion of S has the same order dimension as does S itself. [19] In the category of partially ordered sets and monotonic functions between partially ordered sets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of S is the injective hull of S. [20]