On classes of symmetric and asymmetric concrete logics

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.