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Foundations of finitely supported st...

Alexandru, Andrei.

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Foundations of finitely supported structures = a set theoretical viewpoint /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Foundations of finitely supported structures/ by Andrei Alexandru, Gabriel Ciobanu.
Reminder of title:	a set theoretical viewpoint /
Author:	Alexandru, Andrei.
other author:	Ciobanu, Gabriel.
Published:	Cham :Springer International Publishing : : 2020.,
Description:	xi, 204 p. :ill., digital ;24 cm.
[NT 15003449]:	The World of Structures with Finite Supports -- Finitely Supported Sets: Formal Results -- Choice Principles in Finitely Supported Mathematics -- Connections with Tarski's Concept of Logicality -- Partially Ordered Sets in Finitely Supported Mathematics -- Lattices in Finitely Supported Mathematics -- Constructions of Lattices in Finitely Supported Mathematics -- Galois Connections in Finitely Supported Mathematics -- Several Forms of Infinity in Finitely Supported Mathematics -- Properties of Atoms in Finitely Supported Mathematics -- Freshness in Finitely Supported Mathematics -- Abstraction in Finitely Supported Mathematics -- Relaxing the Finite Support Requirement.
Contained By:	Springer Nature eBook
Subject:	Set theory. -
Online resource:	https://doi.org/10.1007/978-3-030-52962-8
ISBN:	9783030529628

Foundations of finitely supported structures = a set theoretical viewpoint /
Alexandru, Andrei.

Foundations of finitely supported structuresa set theoretical viewpoint /[electronic resource] :by Andrei Alexandru, Gabriel Ciobanu. - Cham :Springer International Publishing :2020. - xi, 204 p. :ill., digital ;24 cm.

The World of Structures with Finite Supports -- Finitely Supported Sets: Formal Results -- Choice Principles in Finitely Supported Mathematics -- Connections with Tarski's Concept of Logicality -- Partially Ordered Sets in Finitely Supported Mathematics -- Lattices in Finitely Supported Mathematics -- Constructions of Lattices in Finitely Supported Mathematics -- Galois Connections in Finitely Supported Mathematics -- Several Forms of Infinity in Finitely Supported Mathematics -- Properties of Atoms in Finitely Supported Mathematics -- Freshness in Finitely Supported Mathematics -- Abstraction in Finitely Supported Mathematics -- Relaxing the Finite Support Requirement.

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-fuzzy subsets 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.

ISBN: 9783030529628

Standard No.: 10.1007/978-3-030-52962-8doiSubjects--Topical Terms:

520081
Set theory.

LC Class. No.: QA248

Dewey Class. No.: 511.322

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100 1 $a Alexandru, Andrei. $3 2203179
245 1 0 $a Foundations of finitely supported structures $h [electronic resource] : $b a set theoretical viewpoint / $c by Andrei Alexandru, Gabriel Ciobanu.
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300 $a xi, 204 p. : $b ill., digital ; $c 24 cm.
505 0 $a The World of Structures with Finite Supports -- Finitely Supported Sets: Formal Results -- Choice Principles in Finitely Supported Mathematics -- Connections with Tarski's Concept of Logicality -- Partially Ordered Sets in Finitely Supported Mathematics -- Lattices in Finitely Supported Mathematics -- Constructions of Lattices in Finitely Supported Mathematics -- Galois Connections in Finitely Supported Mathematics -- Several Forms of Infinity in Finitely Supported Mathematics -- Properties of Atoms in Finitely Supported Mathematics -- Freshness in Finitely Supported Mathematics -- Abstraction in Finitely Supported Mathematics -- Relaxing the Finite Support Requirement.
520 $a 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-fuzzy subsets 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.
650 0 $a Set theory. $3 520081
650 0 $a Finite element method. $3 579714
650 1 4 $a Theory of Computation. $3 892514
650 2 4 $a Mathematics of Computing. $3 891213
650 2 4 $a Mathematical Logic and Foundations. $3 892656
700 1 $a Ciobanu, Gabriel. $3 1070810
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856 4 0 $u https://doi.org/10.1007/978-3-030-52962-8
950 $a Computer Science (SpringerNature-11645)