Overview
- Presents a set theoretical development for the foundations of the theory of finitely supported sets and structures
- Authors collect various results on topic and present them in a uniform manner
- Valuable for researchers in mathematical logic and computer science logic
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About this book
This book presents a set theoretical development for the foundations of the theory of atomic and finitely supported structures. It analyzes whether a classical result can be adequately reformulated by replacing a 'non-atomic structure' with an 'atomic, finitely supported structure’. It also presents many specific properties, such as finiteness, cardinality, connectivity, fixed point, order and uniformity, of finitely supported atomic structures that do not have non-atomic correspondents.
In the framework of finitely supported sets, the authors analyze the consistency of various forms of choice and related results. They introduce and study the notion of 'cardinality' by presenting various order and arithmetic properties. Finitely supported partially ordered sets, chain complete sets, lattices and Galois connections are studied, and new fixed point, calculability and approximation properties are presented. In this framework, the authors study the finitely supported L-fuzzysubsets of a finitely supported set and the finitely supported fuzzy subgroups of a finitely supported group. Several pairwise non-equivalent definitions for the notion of 'infinity' (Dedekind infinity, Mostowski infinity, Kuratowski infinity, Tarski infinity, ascending infinity) are introduced, compared and studied in the new framework. Relevant examples of sets that satisfy some forms of infinity while not satisfying others are provided. Uniformly supported sets are analyzed, and certain surprising properties are presented. Finally, some variations of the finite support requirement are discussed.
The book will be of value to researchers in the foundations of set theory, algebra and logic.
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Table of contents (14 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
Bibliographic Information
Book Title: Foundations of Finitely Supported Structures
Book Subtitle: A Set Theoretical Viewpoint
Authors: Andrei Alexandru, Gabriel Ciobanu
DOI: https://doi.org/10.1007/978-3-030-52962-8
Publisher: Springer Cham
eBook Packages: Computer Science, Computer Science (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-52961-1Published: 21 July 2020
Softcover ISBN: 978-3-030-52964-2Published: 22 July 2021
eBook ISBN: 978-3-030-52962-8Published: 20 July 2020
Edition Number: 1
Number of Pages: XI, 204
Number of Illustrations: 2 b/w illustrations
Topics: Theory of Computation, Mathematics of Computing, Mathematical Logic and Foundations