Covering in the lattice of equational theories and some properties of term finite theories

For countable languages, we completely describe those cardinals κ such that there is an equational theory which covers exactly κ other equational theories. For this task understanding term finite theories is helpful. A theoryT isterm finite provided {ψ:Tτϕ≈ψ} is finite for all terms ϕ. We develop here some fundamental properties of term finite theories and use them, together with Ramsey's Theorem, to prove that any finitely based term finite theory covers only finitely many others. We also show that every term finite theory possesses an independent base and that there are such theories whose pairwise joins are not term finite. The paper was written with the support of NSF Grant MCS-80-01778. Presented by B. Jónsson. Received July 22, 1980. Accepted for publication in final form March 19, 1981.