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 algebra $𝒜=(A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta ,\Phi \in \text{Con}𝒜$ we have $\Theta =\Phi$ whenever ${\left[g\left(a\right)\right]}_{\Theta }={\left[g\left(a\right)\right]}_{\Phi }$ for each $a\in A$. We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset $C\subseteq A$ is a class of some congruence on $\Theta$ containing $g\left(a\right)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).

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...

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra $P$ are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra $P$ is a union of generalized MV-effect algebras and...

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.

The notion of a construction of a fuzzy preference structures is introduced. The properties of a certain class of generated fuzzy implications are studied. The main topic in this paper is investigation of the construction of the monotone generator triplet $(p,i,j)$, which is the producer of fuzzy preference structures. Some properties of mentioned monotone generator triplet are investigated.