Boolean-Valued Universe as an Algebraic System. I: Basic Principles

Gutman, A. E.

doi:10.1134/s0037446619050057

Cited by 3 publications

(3 citation statements)

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“…References are also provided for the author's publications, with the following exception. The present article continues, develops, and supersedes the study initiated in [15][16][17]. The content of those three papers has been redesigned, corrected, extended, and distributed across the new subsections.…”

Section: Discussionmentioning

confidence: 84%

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Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

Gutman

2021

Mathematics

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This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Booleanvalued system. A close interrelation is established between such a universe and the intensional hierarchy. We prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe.

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Section: Discussionmentioning

confidence: 84%

“…(Moreover, [15] is available only in Russian.) Therefore, we confine ourselves to including [15][16][17] in the list of references.…”

Section: Discussionmentioning

confidence: 99%

Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

Gutman

2021

Mathematics

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“…It is also an effective method to measure the centrality of complex networks [5, 6], extended atomic branching [23] and the carbon‐atom skeleton [24]. In full analogy with Estrada index, Fath‐Tabar et al [7] proposed the Laplacian Estrada index LEE , but the difference between SLEE and LEE is chemistry Fullerenes, fluoranthracene and other non‐alternating conjugate species [24–28]. Therefore, it is necessary to study SLEE of graphs.…”

Section: Resultsmentioning

confidence: 99%

On the signless Laplacian Estrada index of uniform hypergraphs

Xue

Zhu

2020

Int J of Quantum Chemistry

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Let H = (V, E) be a hypergraph and B its incidence matrix. Let Q(H) = BBT be the signless Laplacian matrix of H and λ1(Q), λ2(Q), …, λn(Q) are its eigenvalues. The signless Laplacian Estrada index of H is defined as italicSLEE()H=∑i=1neλi()Q which is first extended to hypergraph. We obtain lower and upper bounds for the index in terms of the number of vertices and edges of H. We also determine the unique graph with maximum SLEE among all k‐uniform hypergraphs. In addition, we characterize the extremal hypertrees with the smallest and the largest SLEE among all k‐uniform hypertrees.

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Boolean-Valued Universe as an Algebraic System. II: Intensional Hierarchies

Gutman

2020

Sib Math J

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Boolean-Valued Universe as an Algebraic System. I: Basic Principles

Cited by 3 publications

References 1 publication

Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

On the signless Laplacian Estrada index of uniform hypergraphs

Boolean-Valued Universe as an Algebraic System. II: Intensional Hierarchies

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