Reduced products, Horn sentences, and decision problems
Fred Galvin · Bulletin of the American Mathematical Society · 1967
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).THEOREM 1.A sentence is preserved by proper reduced products iff it is equivalent to a Horn sentence.