In this work we prove that the set of congruences on an nd-groupoid under suitable conditions is a complete lattice which is a sublattice of the lattice of equivalence relations on the nd-groupoid. The study of these conditions allowed to construct a counterexample to the statement that the set of (fuzzy) congruences on a hypergroupoid is a complete lattice.

Every Dedekind complete Riesz space X has a unique sup-completion s , which is lattice cone. This paper aims to present systematic study this cone by extending several known results the general setting, proving new and, in particular, introducing for elements of finite and infinite parts. enables us get satisfactory abstract formulation some classical setting spaces. We prove, version Borel-Can...

In this paper, we give a short proof of the following result of G. Grr atzer and E. T. Schmidt: every m-algebraic lattice can be represented as the lattice of m-complete congruence relations of some m-complete modular lattice.

Obviously each lattice that satisfies either (JID) or (MID) is distributive. A classic result in lattice theory is Funayama’s theorem [5] stating that there is an embedding e of L into a complete Boolean algebra B that preserves all existing joins and meets in L iff L satisfies both (JID) and (MID). Funayama’s original proof was quite involved. For complete L, Grätzer [6, Sec. II.4] gave a more...