On classes of symmetric and asymmetric concrete logics
https://doi.org/10.26907/2949-3919.2024.1.16-30
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.
Keywords
About the Authors
A. M. Bikchentaev
Kazan Federal University, Volga Region Mathematical Center
Russian Federation
Russian Federation
Airat Midkhatovich Bikchentaev
18 Kremlyovskaya str., Kazan 420008
M. Mohamed Ali
Kazan Federal University, Volga Region Mathematical Center
Russian Federation
Russian Federation
Muntadher Mohamed Ali
18 Kremlyovskaya str., Kazan 420008
K. Fawwaz
Kazan Federal University, Volga Region Mathematical Center
Russian Federation
Russian Federation
Khattab Fawwaz
18 Kremlyovskaya str., Kazan 420008
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For citations:
Bikchentaev A.M., Mohamed Ali M., Fawwaz K. On classes of symmetric and asymmetric concrete logics. Mathematics and Theoretical Computer Science. 2024;2(1):16-30. (In Russ.) https://doi.org/10.26907/2949-3919.2024.1.16-30
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