Some non trivial implications between congruence identities
by
Paolo Lipparini
II Universita' di Roma (Tor Vergata)

If I and I' are identities in the language of lattices, then I |=I' means that every lattice satisfying I also satisfies I'; and I |=_c I' means that, whenever V is a variety of algebras, and all congruence lattices of algebras in V satisfy I , then all congruence lattices of algebras in V satisfy I'.

J.B.Nation made the surprising discovery that there are identities I and I' for which I |=_c I' holds, but I |=I' fails. Subsequently, many other such pairs of identities have been found , but all such ``non-trivial congruence implications'' regarded identities stronger than or equivalent (modulo |=_c ) to modularity.

We present a new kind of non-trivial congruence implications; and we get identities that are strictly weaker (even modulo |=_c ) than modularity.

THEOREM. I |=_c I' holds, while I | = I' and I |=_cmod fail, where mod denotes the modular identity, I is

a(b+a(g+ab)) <= g+a(b+ag) ,

and I' is the identity stating that, for every a, b, g , the interval lattice between a(g+a(b+ag)) and a(b+g) is modular.

We also discuss the possibility of finding other congruence implications of a different nature.

Date received: June 13, 1999