Horn sentences excluding a prime

Abstract

A sentence in the language L for the empty type (so the unique relation symbol is equality) must express only properties concerning the cardinality of structures. A sentence Q in L is said to exclude a finite set S of positive integers when the cardinalities of the finite models of Q are exactly the positive integers in the complement S of ~r When S is closed under multiplication and contains 1, such a sentence has to be logically equivalent to a strict Horn sentence [Gr 1979]. K. I. Appel [Ap 1959] determined for any S as above a strict Horn V::l-sentence in L excluding ~r Its size grows superexponentially with the maximum m of S. In the paper [Tu 1984] a polynomial-sized sentence Hp excluding S was found in the case that S consists of a single prime number p. In fact, Hp has size O(p 5 logp). Later also a solution of size O(m m log m), when rn is the maximum of ~r to the general problem was given [Ma-Tu 1985]. This general solution uses discriminator theory to determine equations which have as spectrum the closure under multiplication of the spectrum of a given first order sentence. This useful technique goes back to R. McKenzie [McK 1975]. The particular solution Hp, which excludes the prime p, was worked out [Tu 1984] by using abelian groups instead of discriminator algebras. Here, using the simpler structure of a set with a permutation, we are able to reduce the size of Horn sentences excluding a prime p to O(p 3 log p). We consider the variables {xi:O<-i<p}, {t~:O<_j<p,O<r<_p} and we denote such lists by ~ and t, respectively. Then, we define the following strict Horn quantifier-free formulas in the empty type. All indices, compatibly with the restrictions imposed, range from 0 to p - 1.