On Finite-element Simple Extensions of a Countable Collection of Countable Groupoids

Sin-Min Lee

Belkin and Gorbunov [2] showed that any two finite groupoids can be imbedded into a finite simple groupoid. We prove here a stronger result: Any countable collection $\{A_i\}_{i\in I}$ of countable grupoids can be embedded into a simple groupoid $K(\bigcup_{i\in I} A_i)$ such that $K(\bigcup_{i\in I} A_i)-\bigcup_{i\in I}A_i$ contains only a single element which generates the whole groupoid.