Superextensions of three-element semigroups
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1
Abstract
A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\em upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\em linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\em maximal linked} if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The {\em superextension} $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $\circ: \lambda(X)\times\lambda(X)\to\lambda(X)$ by the formula $\mathcal L\circ\mathcal M=\Big\langle\bigcup_{a\in L}a*M_a:L\in\mathcal L,\;\{M_a\}_{a\in L}\subset\mathcal M\Big\rangle$ for maximal linked upfamilies $\mathcal{L}, \mathcal{M}\in\lambda(X)$. In the paper we describe superextensions of all three-element semigroups up to isomorphism.
Authors and Affiliations
V. Gavrylkiv
Faithful group actions and Schreier graphs
Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical pro...
Lateral continuity and orthogonally additive operators
We generalize the notion of a laterally convergent net from increasing nets to general ones and study the corresponding lateral continuity of maps. The main result asserts that, the lateral continuity of an orthogonally...
Convergence criterion for branched continued fractions of the special form with positive elements
In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are a...
On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends
In the current paper, in the domain D={(t,x):t∈(0,T),x∈(0,L)} we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam utt(t,x)+a2uxxxx(t,x)+buxx(t,x)+cu(t,x)=0, where a,b,c∈R,...
Coupled fixed point results on metric spaces defined by binary operations
In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, unifo...