σ-ring and σ-algebra of Sets1

• Autorzy: Noboru Endou , Kazuhisa Nakasho , Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18], respectively. In the second section, definitions of a ring and a σ-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a σ-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between σ-ring and σ-algebra of sets are formalized and some related examples are given. The formalization is based on [15], and also referred to [9] and [16].

Słowa kluczowe

EN

semiring of sets; σ-ring of sets; σ-algebra of sets

Kategorie tematyczne

• 03E30: Axiomatics of classical set theory and its fragments
• 03B35: Mechanization of proofs and logical operations
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2015
• Tom: 23
• Numer: 1
• Strony: 51-57

Daty

wydano

2015-03-01

otrzymano

2015-02-18

online

2015-03-31

Twórcy

autor

Noboru Endou

• Gifu National College of Technology, Gifu, Japan

autor

Kazuhisa Nakasho

• Shinshu University, Nagano, Japan

autor

Yasunari Shidama

• Shinshu University, Nagano, Japan

Bibliografia

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Identyfikatory

DOI

10.2478/forma-2015-0004

bwmeta1.id-class.MML

SRINGS 3