A Counterexample to the Finite Height Conjecture
نویسنده
چکیده
By J onsson's Lemma, the variety V(K) generated by a nite lattice has only nitely many subvarieties. This led to the conjecture that, conversely, if a lattice variety has only nitely many subvarieties, then it is generated by a nite lattice. A stronger form of the conjecture states that a nitely generated variety V(K) has only nitely many covers in the lattice of lattice varieties, and that each of these is also generated by a nite lattice. Initial investigations near the bottom of supported these conjectures (see [2]{[15]). However, we will show that they are false by constructing an in nite subdirectly irreducible lattice L and a nite lattice F such that V(L) V(F). A modi cation of this construction yields a nite lattice which generates a variety with in nitely many nitely generated covers. The idea for this example goes back to a discussion with Ralph Freese for a joint paper with Ralph, Mick Adams and J urg Schmid [1].
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