Abstract
The purpose of this note is to present a theorem which characterizes, exhibits, and counts the number of idempotents in a symmetric groupoid. The proof is simple and it simplifies similar results in two papers by Harris in which the number of idempotents is counted by graph theoretic terminology and by generating functions. Some immediate corollaries give other combinatorial results in semigroup theory as well as identifying subgroups of the symmetric semigroup.
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