Categories and General Algebraic Structures with Applications (Jan 2017)
The -super socle of the ring of continuous functions
Abstract
The concept of λ-super socle of C(X), denoted by Sλ(X) (i.e., the set of elements of C(X) such that the cardinality of their cozerosets are less than λ, where λ is a regular cardinal number with λ≤|X|) is introduced and studied. Using this concept we extend some of the basic results concerning SCF(X), the super socle of C(X) to Sλ(X), where λ≥ℵ0. In particular, we determine spaces X for which SCF(X) and Sλ(X) coincide. The one-point λ-compactification of a discrete space is algebraically characterized via the concept of λ-super socle. In fact we show that X is the one-point λ-compactification of a discrete space Y if and only if Sλ(X) is a regular ideal and Sλ(X)=Ox, for some x∈X. .
