Concerning a class of permutable congruence relations on loops

Distributive Congruence Lattices of Congruence-permutable Algebras

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

On a class of meaningful permutable laws

The permutability equation G(G(x, y), z) = G(G(x, z), y) is satisfied by many scientific and geometric laws. A few examples among many are: The LorentzFitzGerald Contraction, Beer’s Law, the Pythagorean Theorem, and the formula for computing the volume of a cylinder. If we required that a permutable law be meaningful, the possible forms of a law are considerably restricted. The class of example...

A CONGRUENCE IDENTITY SATISFIED BY m-PERMUTABLE VARIETIES

We present a new and useful congruence identity satisfied by m-permutable varieties. It has been proved in [L1] that every m-permutable variety satisfies a non-trivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every m-permutable variety satisfies the congruence identity αβh = αγh, for h = m[ m+1 2 ]− 1 Here, [ ] denotes int...

Congruence Relations on Multilattices

Principal and Syntactic Congruences in Congruence-distributive and Congruence-permutable Varieties

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite. 2000 Mathematics subject classification: 08B10.