ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 2 (2003)
p. 147 – 157

On Subgroupoid Lattices of some Finite Groupoid
K. Pioro

Abstract.  We investigate finite commutative groupoids $\mathcal{G}=\langle G,\circ\rangle$ such that $g\circ h\neq g$ for all elements $g,h$ of $\mathcal{G}$. First, we show that for any such groupoid, its weak (i.e. partial) subgroupoid lattice uniquely determines its subgroupoid lattice. Next, we characterize the lattice of all weak subgroupoids of such a groupoid. This is a distributive finite lattice satisfying some combinatorial conditions concerning its atoms and join--irreducible elements.

AMS subject classification:  05C65, 05C99, 08A30, 08A55, 20N02, Secondary::05C90, 06B05, 08A05, 08A62
Keywords:  Finite groupoid, partial groupoid, partial and total subgroupoids, subgroupoid lattices, hypergraph, functional directed graph