On Subgroupoid Lattices of Some Finite Groupoid

We investigate finite commutative groupoids G = 〈G, ◦〉 such that g ◦ h 6= g for all elements g, h of 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. In [5] we proved that for any (total) locally finite unary algebra of finite type (i.e. with finitely many unary operations), its weak subalgebra lattice uniquely determines its strong subalgebra lattice. Here we generalize this result for some finite commutative groupoids. Next, in the second part of this paper, necessary and sufficient conditions are found for a lattice to be isomorphic to the weak subgroupoid lattice of such a groupoid. The classical subgroupoids are sometimes called strong as opposed to the other kind of partial subgroupoids, called weak, considered in this paper. Recall that a partial groupoid H is a weak subgroupoid of a (partial) groupoid G iff the carrier of H is contained in the carrier of G, and for any elements g, h of H, if the product g ◦ h is defined in H, then this is also define in G and these two products are equal. The lattices of all weak and strong subgroupoids of a groupoid G are denoted by Sw(G) and Ss(G), respectively. (More details on various kinds of partial subalgebras and lattices of such subalgebras can be found e.g. in [3] or [4]; see also [6]). Theorem 1. Let G = 〈G, ◦〉 be a (total) finite and commutative groupoid such that (∗) g ◦ h 6= g for each g, h ∈ G. Let H = 〈H, ◦〉 be a partial commutative groupoid such that