Many-valued generalizations of two finite intervals in Post's lattice

Abstract

E.L. Post's study (1941) shows that, although the lattice of clones in 2-valued logic is countably infinite, there exist only finitely many clones which contain both constants, and only finitely many which contain the negation function (neg). There are, however, uncountably many k-valued clones for all k>2; in fact, I. Agoston, et al. (1983) have shown that there are uncountably many containing all constants. One may also regard the set of constant functions of two-valued logic as an instance of the set of all noninvertible, unary functions over any finite domain. We show here that, for all k, there are indeed only finitely many clones containing all such functions. We also generalize those clones in Post's lattice which contain neg to the clones containing all permutation functions. Once again, it can be shown that there are only finitely many such clones. The latter result also serves to characterize the homogeneous relation algebras of R. Poschel (1979).<>