$G_{\delta }$-separation axioms are introduced in ordered fuzzy topological spaces and some of their basic properties are investigated besides establishing an analogue of Urysohn’s lemma.

Many forcing notions obtained using the creature technology are naturally connected with certain integer games.

We prove that the quotient algebra P(ℕ)/I over any analytic ideal I on ℕ contains a Hausdorff gap.

We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that ${\kappa }^{<\kappa }=\kappa$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^{\kappa }$ many such spaces up to homeomorphism. We also establish a Kuratowski-like...

We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F:[0,1]\times [0,1]\to [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x,y,z\in ℝ$ such that $x\le y\le z$, it holds that ${\mu }_X\left(y\right)\ge F({\mu }_X\left(x\right),{\mu }_X\left(z\right))$, where ${\mu }_X:ℝ\to [0,1]$ stands here for the membership function...

An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $En{d}_{R}A$ of the R-module ${}_{R}A$, taking any a ∈ A to the right multiplication $a_r\in En{d}_{R}A$ by a, is an isomorphism of algebras. In this case ${}_{R}A$ is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite...

For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_x$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a ${\sum }_2^0$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [${\sum }_1^1\left(X^2\right)]={\sum }_1^1\left(X\right)$ for a wide class of ${\sum }_2^0$-supported σ-ideals.

In a previous work ([3]) we proved that the Nguyen's condition for [f(tilde-A)]α to be equal to f(Aα) also holds for the most general class of the L-fuzzy subsets, where L is an arbitrary lattice. Here we recall the main points of the proof ad present some examples ralated to non-linear lattices.

An overview of generated triangular norms and their applications is presented. Several properties of generated $t$-norms are investigated by means of the corresponding generators, including convergence properties. Some applications are given. An exhaustive list of relevant references is included.

We prove that any countable set of surjective functions on an infinite set of cardinality ℵₙ with n ∈ ℕ can be generated by at most n²/2 + 9n/2 + 7 surjective functions of the same set; and there exist n²/2 + 9n/2 + 7 surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups such as the injective functions, the Baer-Levi semigroups, and the Schützenberger monoids.

In this paper, some generating methods for principal topology are introduced by means of some logical operators such as uninorms and triangular norms and their properties are investigated. Defining a pre-order obtained from the closure operator, the properties of the pre-order are studied.

In this paper we study two ways of generating multi-dimensional aggregation functions. First of all we obtain multi-dimensional OWA operators in two different ways, one of them through quantifiers and the other through sequences. In the first case, we see that all the operators we obtain are multi-dimensional aggregation functions. We then characterize the multi-dimensional aggregation functions that are generated by quantifiers. In the second case, we characterize the sequences that provide multi-dimensional...