Characterizations of the projection bands and some order properties of the lattices of continuous functions

2022/11/21 by Eugene Bilokopytov, Bilokopytov, Eugene · 1 citation
Mathematics · Decision Sciences · #Advanced Topology and Set Theory #Rings, Modules, and Algebras #Fuzzy and Soft Set Theory

paper · pdf · doi:10.48550/arxiv.2211.11192

Abstract

We show that for an ideal H in an Archimedean vector lattice F the following conditions are equivalent: \bullet H is a projection band; \bullet Any collection of mutually disjoint vectors in H, which is order bounded in F, is order bounded in H; \bullet H is an infinite meet-distributive element of the lattice I_F of all ideals in F in the sense that \bigcap_{J∈ J}(H+ J)=H+ \bigcap J, for any J⊂ I_F. Additionally, we show that if F is uniformly complete and H is a uniformly closed principal ideal, then H is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.

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