This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [12]. We introduce n-ary connectives and prove their properties. We derive temporal logic formulas.

We prove that a type-definable Lascar strong type has finite diameter. We also answer some other questions from [1] on Lascar strong types. We give some applications on subgroups of type-definable groups.

We prove that, for any Hausdorff continuum X, if dim X ≥ 2 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then either dim C(X) = 2 or C(X) is not a C-space. This generalizes some results known for metric continua.

By a ternary structure we mean an ordered pair $(U_0,T_0)$, where $U_0$ is a finite nonempty set and $T_0$ is a ternary relation on $U_0$. A ternary structure $(U_0,T_0)$ is called here a directed geodetic structure if there exists a strong digraph $D$ with the properties that $V\left(D\right)=U_0$ and $$T_0(u,v,w)\text{if}\text{and}\text{only}\text{if}{d}_D(u,v)+d_D(v,w)=d_D(u,w)$$ for all $u,v,w\in U_0$, where $d_D$ denotes the (directed) distance function in $D$. It is proved in this paper that there exists no sentence $𝐬$ of the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies...

The Shaffer's definition of the upper and lower expectations of fuzzy variables is considered with respect to randomized fuzzy sets. The notion of randomized fuzzy sets is introduced in order to evaluate fuzzy statistical indices for an arbitrary chosen fuzzy variable. Provided the distribution of the mathematical expectation of a randomized fuzzy variable is known, it is possible to adopt the traditional methods of testing statistical hypotheses for fuzzy variables.We show that this distribution...

Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o.${(P\left(\omega \right)/fin)}^n$, is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).

Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h|D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) in the Borel hierarchy iff K is axiomatized by an infinitary ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective ${\Sigma }_{\alpha }$ or effective ${\Pi }_{\alpha }$ Borel set with a computable infinitary sentence of the same complexity. This result...

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

In this work, we study the relation between the concept lattice of Wille ([5], [6]) and the L-Fuzzy concept lattice ([2]) developed by us. To do it, we have defined an application g that associates to each concept of Wille an L-Fuzzy concept. The main point of this work is to prove that if we are working with a crisp relation between an object set and an attribute set, the concept lattice of Wille is a sublattice of the L-Fuzzy concept lattice. At the end, we show a typical example in the formal...

In the author's 2012 paper, the V-definable Stable Core 𝕊 = (L[S],S) was introduced. It was shown that V is generic over 𝕊 (for 𝕊-definable dense classes), each V-definable club contains an 𝕊-definable club, and the same holds with 𝕊 replaced by (HOD,S), where HOD denotes Gödel's inner model of hereditarily ordinal-definable sets. In the present article we extend this to models of class theory by introducing the V-definable Enriched Stable Core 𝕊* = (L[S*],S*). As an application we obtain...