Semirings of continuous \((0,\infty]\)-valued functions
type: Item
instance of: scholarly article
title: Semirings of continuous \((0,\infty]\)-valued functions
zbMATH Open document ID: 1403.54013
author: E. M. Vechtomov
author: N. V. Shalaginova
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-10-11T00:00:00Z
review text: The authors study the semiring \(C^\infty(X)=C(X,(0,+\infty])\) of all continuous functions defined on a topological space \(X\) with values in the topological semiring \((0,+\infty]\) of all positive real numbers and \(+\infty\) with the order topology. In the first section some basic concepts and notions (semiring, additive absorbing element, ideal, congruence, zero-set, \(H\)-set, \(F\)-spaces, and \(P\)-spaces) are introduced. In Section 2, a general theory of the semirings \(C^\infty(X)\) (\(H\)-ideals and \(H\)-congruence, divisibility, duality, and definability) is considered. In the next sections, properties of the semirings \(C^\infty(X)\) over \(F\)-spaces and \(P\)-spaces are investigated. The last part of the paper is devoted to the description of ideals and congruences in a semiring \(C^\infty(X)\) over a finite discrete space \(X\).
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 46H99
zbMATH DE Number: 6951630
zbMATH Keywords: semiring of continuous functions
zbMATH Keywords: semifield
zbMATH Keywords: \(H\)-ideals
zbMATH Keywords: \(H\)-congruence
zbMATH Keywords: divisibility
zbMATH Keywords: duality
zbMATH Keywords: \(F\)-spaces
zbMATH Keywords: \(P\)-spaces
MaRDI profile type: Publication
full work available at URL: s10958-018-3923-9
OpenAlex ID: W2810985209
cites work: Concerning Rings of Continuous Functions
cites work: Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal
cites work: Q4092470
cites work: Q4700998
cites work: Graphs, dioids and semirings. New models and algorithms.
cites work: Q3666892
cites work: Q4002582
cites work: Applications of the Theory of Boolean Rings to General Topology
cites work: Q3128100
cites work: Q4761609
cites work: Distributive rings of continuous functions and F-spaces
cites work: Q3359376
cites work: Rings of continuous functions. Algebraic aspects
cites work: The determinability of compacts by lattices of ideals and congruences of semirings of continuous \([0, 1]\)-valued functions on them
cites work: Multiplicatively idempotent semirings.
DOI: 10.1007/S10958-018-3923-9
description: scientific article; zbMATH DE number 6951630