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001 978-3-030-52962-8
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020 _a9783030529628
_9978-3-030-52962-8
024 7 _a10.1007/978-3-030-52962-8
_2doi
050 4 _aQA75.5-76.95
072 7 _aUYA
_2bicssc
072 7 _aCOM014000
_2bisacsh
072 7 _aUYA
_2thema
082 0 4 _a004.0151
_223
100 1 _aAlexandru, Andrei.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aFoundations of Finitely Supported Structures
_h[electronic resource] :
_bA Set Theoretical Viewpoint /
_cby Andrei Alexandru, Gabriel Ciobanu.
250 _a1st ed. 2020.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2020.
300 _aXI, 204 p. 2 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _aThe 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 _aThis 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.
650 0 _aComputer science.
650 0 _aComputer science
_xMathematics.
650 0 _aMathematical logic.
650 1 4 _aTheory of Computation.
650 2 4 _aMathematics of Computing.
650 2 4 _aMathematical Logic and Foundations.
700 1 _aCiobanu, Gabriel.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783030529611
776 0 8 _iPrinted edition:
_z9783030529635
776 0 8 _iPrinted edition:
_z9783030529642
856 4 0 _uhttps://doi.org/10.1007/978-3-030-52962-8
912 _aZDB-2-SCS
912 _aZDB-2-SXCS
942 _cSPRINGER
999 _c176525
_d176525