The message below was sent by Boris Schein.


Quoting "Keith A. Kearnes" <kearnes@...>:

> Is the quasivariety of lattices that are embeddable into lattices of
> permuting equivalence relations a variety?
>

Every equivalence relation is a quasi-order (= reflexive and anti-symmetric
relation).

Every lattice is isomorphic to a lattice of permuting quasi-order relations
(the relative product being the join, while the meet is an ordinary
set-theoretical meet.) Moreover, the above isomorphism may be chosen in such
a way that it turns all inf's (all meets of finite or infinite subsets
existing in our lattice) into set-theoretical intersections and satisfies
other restrictive properties.

Of course, every SEMIlattice can be embedded into a semilattice of permuting
equivalence relations.

Boris Schein