Lidia Bazylevych, Dušan Repovš, Michael Zarichnyi
We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.
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In the work it is shown that the space of idempotent probability measures with compact supports is kappa-metrizable if the given Tychonoff space is kappa-metrizable. It is constructed a series of max-plus-convex subfunctors of the functor of idempotent probability measures with compact supports. Further, it is established that the functor of idempotent probability measures with the compact supports preserves openness of continuous maps.
In this paper we introduce a metrics on the space of idempotent probability measures on a given compactum, which extends the metrics on the compactum. It is proven the introduced metrics generates the pointwise convergence topology on the space of idempotent probability measures
In the paper we proved that for a compact $X$ inclusion $I_{f}(X)\in ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor $I_{f}$ preserves property of a compact to be $Q$-manyfold or a Hilbert cube, preserves property of map layers to be $ANR$-compact, compact $Q$-manyfold and a Hilbert cube (the finite sum of Hilbert cubes)
In this paper we construct a metric on the space of idempotent probability measures on the given compactum, which is an idempotent analog of the Kantorovich metric on the space of probability measures.
A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.
We establish some geometrical properties of the space of idempotent probability measures. In particular, for a compact $X$ it is established that if the space $I_{3}(X)\backslash X$ is hereditary normally, then $X$ is metrizable; some subsets allocate of the space of idempotent probability measures which are, respectively, $Z$-sets, $\max$-$\mbox{plus}$-convex subsets, $G_δ$-sets.
The notion of max-min measure is a counterpart of the notion of max-plus measure (Maslov measure or idempotent measure). In this paper we consider the spaces of max-min measures on the compact Hausdorff spaces. It is proved that the obtained functor of max-min measures is isomorphic to the functor of max-plus (idempotent) measures considered by the second-named author. However, it turns out that the monads generated by these functors are not isomorphic.
We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of the existence of invariant idempotent measures for such IFSs.
In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continous mappings is perfect metrisable
In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continuous mappings is perfect metrizable.
The set of all idempotent probability measures (Maslov measures) on a compact Hausdorff space endowed with the weak* topology determines is functorial on the category $\comp$ of compact Hausdorff spaces. We prove that the obtained functor is normal in the sense of E. Shchepin. Also, this functor is the functorial part of a monad on $\comp$. We prove that the idempotent probability measure monad contains the hyperspace monad as its submonad. A counterpart of the notion of Milyutin map is defined for the idempotent probability measures. Using the fact of existence of Milyutin maps we prove that …
Invariant idempotent $\ast$-measures
本文研究紧度量空间上迭代函数系统的不变测度问题，通过将幂等测度推广为适用于任意三角模 $\ast$ 的 $\ast$-幂等测度，证明了相应不变 $\ast$-幂等测度的存在性与唯一性。
We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Mark…
Invariant idempotent $\ast$-measures for generalized iterated function systems
本文研究了广义不变函数系统中$\ast$-测度的不变性，证明了其存在性和唯一性。
Very Schwartz coidempotents and continuous spectrum
本文引入不稳定粉碎谱函子的连续版本，将可对偶化对称幺半稳定可表现 ∞-范畴对应到稳定紧空间，并用新定义的非常 Schwartz 幂等元刻画其开子集，进而证明稳定紧空间上谱层的 Tannaka 对偶。
Topological and differentiable aspects of Clifford semigroups
本文研究了克利福德半群中代数结构、拓扑与可微性的相互作用，通过构建兼容度量、确立极大子群为李群的条件以及证明结构刚性定理来探讨其核心问题。
Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture
本文研究可定义群上的幂等Keisler测度，提出一般传递性并证明其足以发展局部稳定群论，在阿贝尔群及rosy群等新情形确立一般稳定幂等型的一般传递性，分类幂等测度，并构造可数NIP群测度卷积半群的极小左理想，证明修正的Newelski猜想。
A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in t…
The aim of this paper is to prove the existence of inductive and inverse limits of direct and inverse systems in a certain category of compact metric spaces as well as of compact metric groups. Some applications are presented.
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that Hausdorff compact partial metric spaces are metrizable. In the second part of this article we discuss the significance of bottom sets of partial metric spaces in fixed point theorems for mappings acting in these spaces.
We consider a certain metric on the space of all convex compacta in $\R^{n}$, introduced by A. Pliś. The set of strictly convex compacta is a complete metric subspace of the metric space of convex compacta with respect to this metric. We present some applications of this metric to the problems of set-valued analysis, in particular we estimate the distance between two compact sets with respect to this metric and to the Hausdorff metric.
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
Finite approximations of countable metric and ultrametric compacta
通过改编同伦重构定理，证明可数度量或超度量紧空间可由有限 $T_0$ 空间序列的逆极限进行拓扑重构。
The ultrametrization of the set of all probability measures of compact support on the ultrametric spaces was first defined by Hartog and de Vink. In this paper we consider a similar construction for the so called max-min measures on the ultrametric spaces. In particular, we prove that the functors max-min measures and idempotent measures are isomorphic. However, we show that this is not the case for the monads generated by these functors.
It is known that the Stone-Čech compactification of a non-compact metrizable space $X$ is approximated by the collection of Smirnov compactifications of $X$ for all compatible metrics on $X$. We investigate the smallest cardinality of a set $D$ of compatible metrics on the countable discrete space $ω$ such that, the Stone-Čech compactification of $ω$ is approximated by Smirnov compactifications for all metrics in $D$, but any finite subset of $D$ does not suffice. We also study the corresponding cardinality for Higson compactifications.
Let (X,d) be a metric space and m\in X. Suppose that ϕ:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{ϕ,m} on X which is equivalent to d. If d^{ϕ,m} is totally bounded, its completion is a compactification of (X,d). As examples, we construct two compactifications of (\mathhbold{R}^s,d_E), where d_E is the Euclidean metric and s\geq 2.
We show that the direct sum of uncountably many non-Abelian groups does not embed into the group of homeomorphisms of a compact metric space.
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