Properties preserved under homomorphism

The main result of this paper is a characterization of those sentences of the predicate calculus whose validity is preserved under passage from an abstract algebraic system to any homomorphic image of the system. An algebraic system is here construed to be a set together with certain operations and relations, including identity, defined for elements of the set. The sentences under consideration will contain symbols for these operations and relations, and variables whose range is the set of elements of the system, together with the usual logical symbols, but will contain no variables whose range consists of sets, relations, or functions. Such a sentence will be called positive if it contains the logical symbols for conjunction, disjunction and quantification only, but not the symbol for negation. It will be shown that: