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Remarks on the Stone Spaces of the Integers and the Reals without AC

Horst Herrlich, Kyriakos Keremedis, Eleftherios Tachtsis (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2 ( X ) , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

Representation and construction of homogeneous and quasi-homogeneous n -ary aggregation functions

Yong Su, Radko Mesiar (2021)

Kybernetika

Homogeneity, as one type of invariantness, means that an aggregation function is invariant with respect to multiplication by a constant, and quasi-homogeneity, as a relaxed version, reflects the original output as well as the constant. In this paper, we characterize all homogeneous/quasi-homogeneous n -ary aggregation functions and present several methods to generate new homogeneous/quasi-homogeneous n -ary aggregation functions by aggregation of given ones.

Representation of uni-nullnorms and null-uninorms on bounded lattices

Yi-Qun Zhang, Ya-Ming Wang, Hua-Wen Liu (2024)

Kybernetika

In this paper, we present the representation for uni-nullnorms with disjunctive underlying uninorms on bounded lattices. It is shown that our method can cover the representation of nullnorms on bounded lattices and some of existing construction methods for uni-nullnorms on bounded lattices. Illustrative examples are presented simultaneously. In addition, the representation of null-uninorms with conjunctive underlying uninorms on bounded lattices is obtained dually.

Representations of the direct product of matrix algebras

Daniele Guido, Lars Tuset (2001)

Fundamenta Mathematicae

Suppose B is a unital algebra which is an algebraic product of full matrix algebras over an index set X. A bijection is set up between the equivalence classes of irreducible representations of B as operators on a Banach space and the σ-complete ultrafilters on X (Theorem 2.6). Therefore, if X has less than measurable cardinality (e.g. accessible), the equivalence classes of the irreducible representations of B are labeled by points of X, and all representations of B are described (Theorem 3.3).

Resolvability in c.c.c. generic extensions

Lajos Soukup, Adrienne Stanley (2017)

Commentationes Mathematicae Universitatis Carolinae

Every crowded space X is ω -resolvable in the c.c.c. generic extension V Fn ( | X | , 2 ) of the ground model. We investigate what we can say about λ -resolvability in c.c.c. generic extensions for λ > ω . A topological space is monotonically ω 1 -resolvable if there is a function f : X → ω 1 such that { x ∈ X : f ( x ) ≥ α } ⊂ d e n s e X for each α < ω 1 . We show that given a T 1 space X the following statements are equivalent: (1) X is ω 1 -resolvable in some c.c.c. generic extension; (2) X is monotonically ω 1 -resolvable; (3) X is ω 1 -resolvable in the Cohen-generic extension V Fn ( ω 1 , 2 ) ....

Revealed automorphisms

Jiří Sgall, Antonín Sochor (1991)

Commentationes Mathematicae Universitatis Carolinae

We study automorphisms in the alternative set theory. We prove that fully revealed automorphisms are not closed under composition. We also construct some special automorphisms. We generalize the notion of revealment and Sd-class.

Revealments

Antonín Sochor, Petr Vopěnka (1980)

Commentationes Mathematicae Universitatis Carolinae

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