On closure under direct product
Citations Over TimeTop 10% of 1958 papers
Abstract
In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question. We shall discuss relational systems of the form where A and R are non-empty sets; each element of R is an ordered triple 〈 a, b, c 〉, with a, b, c ∈ A . 1 If the triple 〈 a, b, c 〉 belongs to the relation R , we write R ( a, b, c ); if 〈 a, b, c 〉 ∉ R , we write ( a, b, c ). If x 0 , x 1 and x 2 are variables, then R ( x 0 , x 1 , x 2 ) and x 0 = x 1 are predicates . The expressions ( x 0 , x 1 , x 2 ) and x 0 ≠ x 1 will be referred to as negations of predicates . We speak of α 1 , …, α n as terms of the disjunction α 1 ∨ … ∨ α n and as factors of the conjunction α 1 ∧ … ∧ α n . A sentence (open, closed or neither) of the form where each Q i (if there be any) is either the universal or the existential quantifier and each α i, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form .
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