Formulas on the lattice of fuzzy subalgebras in universal algebra
We study what first-order sentences give the equivalence in the meaning that holding on the lattice of subuniverses implies holding on that of fuzzy subalgebras, and vice versa. We introduce the notions of transferable form and strict-transferable form of open formulas, and show that universal sentences whose matrixes are of the former type and existential sentences whose matrixes are of the latter type give the equivalence for every abstract algebra. After that, we study the case when sentences are general. For the purpose, we examine Skolem normal forms of given sentences. (c) 2007 Elsevier B.V. All rights reserved.
fuzzy subalgebra skolem normal form first-order formula