Reduced products, Horn sentences, and decision problems

Abstract

We shall consider a first order language with equality, and the corresponding structures. Reduced products {powers) and Horn sentences are defined in [4, pp. 199, 211]. An V n sentence is a prenex sentence whose prefix consists of a block of universal quantifiers, followed by a block of existential quantifiers, followed by another block of universal quantifiers, and so on n times; a n sentences are defined dually. An EC (HC, V n C) is the class of all models of an elementary sentence (Horn sentence, V n sentence), and an EC A (HCA, VWCA) is the class of all models of a set of elementary sentences (Horn sentences, V n sentences

Boolean powers, recursive models, and the Horn theory of a structure

“…In [20] for each elementary characteristic («,/?, q) of a complete theory of a Boolean algebra and any first-order language L, we defined semantically V ((n, p,q)) the set of valid sentences of L of reduced power structures of type {n 9 p 9 q). Also we gave a syntactic characterization of V({n 9 In [20] ( §3) we demonstrated numerous results about the logics V ((n 9 p, q)) and the more interesting of these results required that L be sufficiently large, i.e., contain the language of Boolean algebras. In this section we intend to show that all of the results of [20] …”

“…φ 0 and let 33, have elementary characteristic (1,1,0). Clearly, 33 0^φ while » o l*iΓ ^^lί^ol* PI (Corollary 2.8) which has elementary characteristic <l,/?,0) by [7], [29], or [23] 9 we do not consider in this paper this alternative. Clearly % f satisfies c λ ^ c 0 as well as VJ^ \fx n (3ί R (x^... ,x n ) -c ι V dc R (x x ,... 9 x n ) = c 0 ) for each R in L. Conversely, any structure © for Zy satisfying these sentences determines a unique 3ί for L such that % f = ©.…”

“…By [7] any elementary characteristic (n 9 p 9 q) determines a complete theory of a Boolean algebra and this complete theory is decidable. Consequently, if 2 T /Dt (n 9 p, q) 9 then there is a recursive model 93 ( (Λ, p, q)) of that elementary characteristic. Given that 9ί is recursive, it is easy to see that 2t[93 ((π, p, q))]* is recursive by definition in [26] and 3C[SB«n, p 9 q))]* = % J /D by [3] (Theorem 4.3(i) and (v)).…”

Theories with the Independence Property

“…Apparently, this is a disjunction of crisp sentences. By [7,Thm. 6], a theory based on such sentences is stable under reduced powers.…”