On classes of symmetric and asymmetric concrete logics-Reference-Cited by-同舟云学术

On classes of symmetric and asymmetric concrete logics

Published:2024-04-15 Issue:1 Volume:2 Page:16-30
ISSN:2949-3919
Container-title:Mathematics and Theoretical Computer Science
language:
Short-container-title:jour

Author:

Bikchentaev A. M.¹,Mohamed Ali M.¹,Fawwaz K.¹

Affiliation:

1. Kazan Federal University, Volga Region Mathematical Center

Abstract

We refined the axiomatics of asymmetric logics. For logics X(km, k) of family subsets of the km-element set X, which cardinal numbers are multiples of k we completely described the cases in which X(km, k) a) is symmetric or b) is asymmetric. For an infinite set Ω and a natural number n ≥ 2 we constructed the concrete logics EΩn and completely described the cases in which these logics are asymmetric. For asymmetric logics E we determine when both the set A ∈ E and its complement Ac are atoms of the logic E. Let a symmetric logic E of a finite set Ω be not a Boolean algebra, and let A be an algebra of subsets from Ω, and assume that E ⊂ A. Then there exists a measure on E, that does not admit an extension to a measure on A.

Publisher

Kazan Federal University

Reference16 articles.

1. A.N. Sherstnev, On Boolean logics, Uchenye Zapiski Kazanskogo Universiteta, 128 (2) Kazan Univ., Kazan, 1968, 48–62 [in Russian]. DOI: https://www.mathnet.ru/eng/uzku/v128/i2/p48

2. S.P. Gudder, Stochastic methods in quantum mechanics, North-Holland Series in Probability and Applied Mathematics. North-Holland, New York–Oxford, 1979.

3. S.P. Gudder, An extension of classical measure theory, SIAM Rev. 26 (1), 71–89 (1984). DOI: https://doi.org/10.1137/1026002

4. G.D. Lugovaya, A.N. Sherstnev, Functional analysis: special courses, Editorial URSS, Moscow, 2019 [in Russian].

5. P.G. Ovchinnikov, Galois correspondence associated with the extension of charges on the σ-class of subsets of a finite set, Russian Math. (Iz. VUZ), 38 (5), 34–38 (1994). URL: https://www.mathnet.ru/eng/ivm/y1994/i5/p36