Semirings of continuous \((0,\infty]\)-valued functions
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 \(X\).
scientific article; zbMATH DE number 6951630
scientific article