Mehran, S., Namdari, M. (2017). The $\lambda$-super socle of the ring of continuous functions. Categories and General Algebraic Structures with Applications, 6(Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)), 37-50.

Simin Mehran; Mehrdad Namdari. "The $\lambda$-super socle of the ring of continuous functions". Categories and General Algebraic Structures with Applications, 6, Speical Issue on the Occasion of Banaschewski's 90th Birthday (I), 2017, 37-50.

Mehran, S., Namdari, M. (2017). 'The $\lambda$-super socle of the ring of continuous functions', Categories and General Algebraic Structures with Applications, 6(Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)), pp. 37-50.

Mehran, S., Namdari, M. The $\lambda$-super socle of the ring of continuous functions. Categories and General Algebraic Structures with Applications, 2017; 6(Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)): 37-50.

The $\lambda$-super socle of the ring of continuous functions

^{}Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Abstract

The concept of $\lambda$-super socle of $C(X)$, denoted by $S_\lambda(X)$ (i.e., the set of elements of $C(X)$ such that the cardinality of their cozerosets are less than $\lambda$, where $\lambda$ is a regular cardinal number with $\lambda\leq |X|$) is introduced and studied. Using this concept we extend some of the basic results concerning $SC_F(X)$, the super socle of $C(X)$ to $S_\lambda(X)$, where $\lambda \geq\aleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_\lambda(X)$ coincide. The one-point $\lambda$-compactification of a discrete space is algebraically characterized via the concept of $\lambda$-super socle. In fact we show that $X$ is the one-point $\lambda$-compactification of a discrete space $Y$ if and only if $S_\lambda(X)$ is a regular ideal and $S_\lambda(X)=O_x$, for some $x\in X$.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

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