Rings of real functions in Pointfree Topology

• Main Author: Gutiérrez García, Javier
• Publication Date: 2010
• Other Authors: Picado, Jorge
• Format: Other
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: https://hdl.handle.net/10316/13708
• topic: Frame; Locale; Sublocale; Frame of reals; Scale; Frame real function; Continuous real function; Lower semicontinuous; Upper semicontinuous; Lattice-ordered ring; Ring of continuous functions in pointfree topology; Strict insertion

This paper deals with the algebra F(L) of real functions of a frame L and its subclasses LSC(L) and USC(L) of, respectively, lower and upper semicontinuous real functions. It is well-known that F(L) is a lattice-ordered ring; this paper presents explicit formulas for its algebraic operations which allow to conclude about their behaviour in LSC(L) and USC(L). As applications, idempotent functions are characterized and the results of [10] about strict insertion of functions are signi cantly improved: general pointfree formulations that correspond exactly to the classical strict insertion results of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces are derived. The paper ends with a brief discussion concerning the frames in which every arbitrary real function on the -dissolution of the frame is continuous