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Foundations of finitely supported st...

Alexandru, Andrei.

Foundations of finitely supported structuresa set theoretical viewpoint /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Foundations of finitely supported structuresby Andrei Alexandru, Gabriel Ciobanu.
其他題名:	a set theoretical viewpoint /
作者:	Alexandru, Andrei.
其他作者:	Ciobanu, Gabriel.
出版者:	Cham :Springer International Publishing :2020.
面頁冊數:	xi, 204 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
標題:	Set theory.
電子資源:	https://doi.org/10.1007/978-3-030-52962-8
ISBN:	9783030529628$q(electronic bk.)

Foundations of finitely supported structuresa 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$q(electronic bk.)

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

190892
Set theory.

LC Class. No.: QA248

Dewey Class. No.: 511.322

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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.
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