Let $X$ be a finite subset of real numbers. Let $G$ be the collection of all non empty subsets of $X$ and $G_{0}$ be the collection of all singleton subsets of $X$.

We define two maps $r,s:G \to G_{0}$ with $$r(A)=\text{The singletone consisting of Maximum} \;\; of\;\; A$$ and $$s(A)=\text{The singleton consisting of minimum}\;\; of \;\; A$$

This provide a semi groupoid structure on the pair $(G,G_{0})$ whose composition law is the union operation $A\circ B= A \cup B$.

What is the precise description of the groupoid associated with this semigroupoid? The resulting groupoid gives us a groupoid $C^{*}$ algebra. Is this (finite dimensional) $C^{*}$ algebra studied, already?What is its precise description?

What about if we replace the pair $(G,G_{0})$ with the collection of compact subsets of the interval and its singleton subsets, respectively?