On Axiomatics of Symmetric and Asymmetric Concrete Logics

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 $$\Omega$$ and a natural number $$n\geq 2$$ we constructed the concrete logics $$\mathcal{E}^{n}_{\Omega}$$ and completely described the cases in which these logics are asymmetric. For asymmetric logics $$\mathcal{E}$$ we determine when both the set $$A\in\mathcal{E}$$ and its complement $$A^{c}$$ are atoms of the logic $$\mathcal{E}$$ . Let a symmetric logic $$\mathcal{E}$$ of a finite set $$\Omega$$ be not a Boolean algebra, and let $$\mathcal{A}$$ be an algebra of subsets from $$\Omega$$ , and assume that $$\mathcal{E}\subset\mathcal{A}$$ . Then there exists a measure on $$\mathcal{E}$$ , that does not admit an extension to a measure on $$\mathcal{A}$$ .