Ebook: Algebras and Orders

In the summer of 1991 the Department of Mathematics and Statistics of the Universite de Montreal was fortunate to host the NATO Advanced Study Institute "Algebras and Orders" as its 30th Seminaire de mathematiques superieures (SMS), a summer school with a long tradition and well-established reputation. This book contains the contributions of the invited speakers. Universal algebra- which established itself only in the 1930's- grew from traditional algebra (e.g., groups, modules, rings and lattices) and logic (e.g., propositional calculus, model theory and the theory of relations). It started by extending results from these fields but by now it is a well-established and dynamic discipline in its own right. One of the objectives of the ASI was to cover a broad spectrum of topics in this field, and to put in evidence the natural links to, and interactions with, boolean algebra, lattice theory, topology, graphs, relations, automata, theoretical computer science and (partial) orders. The theory of orders is a relatively young and vigorous discipline sharing certain topics as well as many researchers and meetings with universal algebra and lattice theory. W. Taylor surveyed the abstract clone theory which formalizes the process of compos­ ing operations (i.e., the formation of term operations) of an algebra as a special category with countably many objects, and leading naturally to the interpretation and equivalence of varieties.

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The book consists of the lectures presented at the NATO ASI on `Algebras and Orders' held in 1991 at the Universit? de Montr?al. The lectures cover a broad spectrum of topics in universal algebra, Boolean algebras, lattices and orders, and their links with graphs, relations, topology and theoretical computer science. More specifically, the contributions deal with the following topics: Abstract clone theory (W. Taylor); Hyperidentities and hypervarieties (D. Schweigert); Arithmetical algebras and varieties (A. Pixley); Boolean algebras with operators (B. Jonsson); Algebraic duality (B. Davey); Model-theoretic aspects of partial algebras (P. Burmeister); Free lattices (R. Freese); Algebraic ordered sets (M. Ern?); Diagrams of orders (I. Rival); Essentially minimal groupoids (H. Machida, I.G. Rosenberg); and Formalization of predicate calculus (I. Fleischer). Most of the papers are up-to-date surveys written by leading researchers, or topics that are either new or have witnessed recent substantial progress. In most cases, the surveys are the first available in the literature. The book is accessible to graduate students and researchers.

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The book consists of the lectures presented at the NATO ASI on `Algebras and Orders' held in 1991 at the Universit? de Montr?al. The lectures cover a broad spectrum of topics in universal algebra, Boolean algebras, lattices and orders, and their links with graphs, relations, topology and theoretical computer science. More specifically, the contributions deal with the following topics: Abstract clone theory (W. Taylor); Hyperidentities and hypervarieties (D. Schweigert); Arithmetical algebras and varieties (A. Pixley); Boolean algebras with operators (B. Jonsson); Algebraic duality (B. Davey); Model-theoretic aspects of partial algebras (P. Burmeister); Free lattices (R. Freese); Algebraic ordered sets (M. Ern?); Diagrams of orders (I. Rival); Essentially minimal groupoids (H. Machida, I.G. Rosenberg); and Formalization of predicate calculus (I. Fleischer). Most of the papers are up-to-date surveys written by leading researchers, or topics that are either new or have witnessed recent substantial progress. In most cases, the surveys are the first available in the literature. The book is accessible to graduate students and researchers.
Content:
Front Matter....Pages i-xvii
Partial Algebras — An Introductory Survey....Pages 1-70
Duality Theory on Ten Dollars a Day....Pages 71-111
Algebraic Ordered Sets and Their Generalizations....Pages 113-192
A Boolean Formalization of Predicate Calculus....Pages 193-198
Lectures on Free Lattices....Pages 199-237
A Survey of Boolean Algebras with Operators....Pages 239-286
Essentially Minimal Groupoids....Pages 287-316
Functional and Affine Completeness and Arithmetical Varieties....Pages 317-357
Reading, Drawing, and Order....Pages 359-404
Hyperidentities....Pages 405-506
Abstract Clone Theory....Pages 507-530
Back Matter....Pages 531-553

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The book consists of the lectures presented at the NATO ASI on `Algebras and Orders' held in 1991 at the Universit? de Montr?al. The lectures cover a broad spectrum of topics in universal algebra, Boolean algebras, lattices and orders, and their links with graphs, relations, topology and theoretical computer science. More specifically, the contributions deal with the following topics: Abstract clone theory (W. Taylor); Hyperidentities and hypervarieties (D. Schweigert); Arithmetical algebras and varieties (A. Pixley); Boolean algebras with operators (B. Jonsson); Algebraic duality (B. Davey); Model-theoretic aspects of partial algebras (P. Burmeister); Free lattices (R. Freese); Algebraic ordered sets (M. Ern?); Diagrams of orders (I. Rival); Essentially minimal groupoids (H. Machida, I.G. Rosenberg); and Formalization of predicate calculus (I. Fleischer). Most of the papers are up-to-date surveys written by leading researchers, or topics that are either new or have witnessed recent substantial progress. In most cases, the surveys are the first available in the literature. The book is accessible to graduate students and researchers.
Content:
Front Matter....Pages i-xvii
Partial Algebras — An Introductory Survey....Pages 1-70
Duality Theory on Ten Dollars a Day....Pages 71-111
Algebraic Ordered Sets and Their Generalizations....Pages 113-192
A Boolean Formalization of Predicate Calculus....Pages 193-198
Lectures on Free Lattices....Pages 199-237
A Survey of Boolean Algebras with Operators....Pages 239-286
Essentially Minimal Groupoids....Pages 287-316
Functional and Affine Completeness and Arithmetical Varieties....Pages 317-357
Reading, Drawing, and Order....Pages 359-404
Hyperidentities....Pages 405-506
Abstract Clone Theory....Pages 507-530
Back Matter....Pages 531-553
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