Weak subgroupoid lattices. (Q2877062)

Let \(\mathcal A=(A,(f^A_t)_{t\in T})\) and \(\mathcal B=(B,(f^B_t)_{t\in T})\) be partial algebras of the same type. Then \(\mathcal B\) is called a weak subalgebra of \(\mathcal A\) if \(B\subseteq A\) and \(f^B_t\subseteq f^A_t\) for all \(t\in T\). The set \(S_w(\mathcal A)\) of all weak subalgebras of \(\mathcal A\) is an algebraic and distributive lattice. The complete characterization of the weak subalgebra lattices of partial algebras is given by \textit{W. Bartol} [Commentat. Math. Univ. Carol. 31, No. 3, 405--410 (1990; Zbl 0711.08006)]. But this characterization does not yield any information about the type of algebra.NEWLINENEWLINE The first main theorem of this paper characterizes the weak subgroupoid lattices of partial groupoids. More precisely, a lattice \(\mathcal L\) is isomorphic to the weak subgroupoid lattice \(S_w(\mathcal G)\) for some partial groupoid \(\mathcal G\) if and only if \(\mathcal L\) is algebraic and distributive, each of its elements is a join of completely join-irreducible elements, every non-zero and non-atomic completely join-irreducible element covers either an atom or a join of a pair of atoms or a join of a triple of atoms and additionally, a simple combinatorial condition concerning their atoms and join-irreducible elements is satisfied.NEWLINENEWLINE The second main theorem shows that for each finite total groupoid \(\mathcal G\), its weak subgroupoid lattice \(S_w(\mathcal G)\) uniquely determines its subgroupoid lattice \(S(\mathcal G)\). Next, a simple modification of the proof gives the following result: If \(\mathcal A\) and \(\mathcal B\) are finite partial algebras of the same finite type such that \(\mathcal A\) is total and weak subalgebra lattices \(S_w(\mathcal A)\), \(S_w(\mathcal B)\) are isomorphic, then \(\mathcal B\) is also total and subalgebra lattices \(S(\mathcal A)\), \(S(\mathcal B)\) are isomorphic (moreover, if \(f\) is an isomorphism from \(S_w(\mathcal A)\) to \(S_w(\mathcal B)\), then \(f\) restricted to \(S(\mathcal A)\) is an isomorphism between \(S(\mathcal A)\) and \(S(\mathcal B)\)).NEWLINENEWLINE Particular cases of the above two results are proved by \textit{K. Pióro} [Acta Math. Univ. Comen., New Ser. 72, No. 2, 147--157 (2003; Zbl 1087.08003)].