A characterization of identities implying congruence modularity. I

Alan Day, Ralph Freese
1980 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
0. Introduction. In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras J^ satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jônsson showed in [10] that from this "congruence modularity" of a variety of algebras one can even deduce the (stronger) Arguesian identity. These and similar results [3; 5; 9; 12;
more » ... sults [3; 5; 9; 12; 18; 21] induced jônsson in [17; 18] to introduce the following notions. For a variety of algebras J^, Con(jT) = HSPG(JT) is the (congruence) variety of lattices generated by the class B(J^) of all congruence lattices 9(^4), A C jtf. Secondly if e is a lattice identity, and 2 is a set of such, 2 t= c e holds if for any variety Jf, Con(jT) != 2 implies Con(jT) t e. In [2] and [16] characterizations of Con(J^) t= mod and Con(J^) t dist were found (mod (dist) is the modular (resp. distributive) law). These statements express the so-called congruence modularity or congruence distributivity of a variety J^. Furthermore in [11] it was shown that for a variety of semigroups J^, Con(J^) t= e where e is any non-trivial lattice identity implies J^ is congruence modular. The aforementioned results led to a conjecture that there existed no proper non-modular congruence varieties but this conjecture was shattered by a recent result of Polin [25], where a variety of algebras SP is produced that is not congruence modular and which has Con(^) 7 e ££ . A detailed analysis of this variety &P (and Con(^)) has allowed us to produce several complete characterizations of congruence modularity and to answer some related questions about congruence varieties and the congruence satisfaction relation 1= c . The main result (6.1) states that Con(^) is the smallest non-modular congruence variety (of lattices). The proof of this fact involves showing that the lattices, 6(Fp(n)) {n < w), are in fact splitting lattices with conjugate splitting equations f w . These results allow us to prove a very strong compactness result that 2 t= c mod if and only if ô t= c mod for some 5 Ç 2. The splitting equations allow us to characterize "Ô t c mod" in
doi:10.4153/cjm-1980-087-6 fatcat:7vosfbkpdbfrhkgy2jm3f77zti