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We address some phenomena about the interaction between lower semicontinuous submeasures on N and F σ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological F σ ideals. We give a partial answers to the question of whether every nonpathological tall F σ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological F σ ideals using sequences in Banach spaces. [ABSTRACT FROM AUTHOR]

Several tasks in artificial intelligence require the ability to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper, we exploit the algebraic framework of mathematical morphology in the context of propositional logic and define operations such as dilation or erosion of a set of formulas. We derive concrete operators, based on a semantic approach, that have an intuitive interpretation and that are formally well behaved, to perform revision, fusion and abduction. Computation and tractability are addressed, and simple examples illustrate the main results. [ABSTRACT FROM AUTHOR]

We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse sub-semigroups of the symmetric inverse semigroup I(N) of all partial bi-jections between subsets of N. We also study whether our examples satisfy automatic continuity. [ABSTRACT FROM AUTHOR]

We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} on the semigroup I ⁢ (X) {I(X)} of partial bijections of X, with 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism θ : G → S {\theta:G\to S} , where S is an inverse monoid, can be extended to a semigroup homomorphism θ * : T → S {\theta^{*}:T\to S} for any inverse semigroup T with 풮 ⁡ (G) ⊆ T ⊆ P * ⁢ (G) × G {\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G} , with P * ⁢ (G) {P^{*}(G)} being the semigroup of non-empty subsets of G, and such that E ⁢ (S) {E(S)} satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Γ ⁢ (X) {\Gamma(X)} , the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X. [ABSTRACT FROM AUTHOR]

Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) (X, f) is equicontinuous, (2) (X, f) is distal and (3) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group. [ABSTRACT FROM AUTHOR]

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Given a family C of infinite subsets of N , we study when there is a Borel function S : 2N to 2N such that for every infinite x ∈ 2N, S(x) ∈ C and S(x) ⊆ x. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams' theorem admits such a Borel selector. However, we also show that the analogous result for Galvin's lemma is not true by proving that there is an Fσ tall ideal on N without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a Π12 tall ideal on N without a tall closed subset. [ABSTRACT FROM AUTHOR]

We show that the following properties are preserved under inverse limits: countable fan-tightness, q+, discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on X. If G and X are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups. [ABSTRACT FROM AUTHOR]

Given a continuum X, for each A ⊆ X, the Jones' set function T is defined by T(A) = {x ∊ X : for each subcontinuum K such that x ∊ Int(K), then K N A = 0}. We show that D = {T({x}) : x ∊ X} is a decomposition of X when T is continuous (restricted to the hyperspace 2X). We present a characterization of the continuity of T and answer several open questions posed by D. Bellamy. [ABSTRACT FROM AUTHOR]

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a coideal H is semiselective if and only if Matet's game-theoretic characterization of the H-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory, and gives a game-theoretic counterpart to a theorem of Farah, asserting that a co-ideal H is semiselective if and only if the family of H-Ramsey subsets of N[∞] coincides with the family of those sets having the abstract H-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H all sets of real numbers are H-Ramsey. [ABSTRACT FROM AUTHOR]