Implicative filters of lattice implication algebras. (English) Zbl 0876.03035

In order to search for the logical system whose propositional value is given in a lattice, Y. Xu [J. Southwest Jiaotong Univ. 1993, No. 1, 20-27 (1993; Zbl 0784.03035)] proposed the concept of lattice implication algebras, and discussed some of their properties. Y. Xu and K. Qin [J. Fuzzy Math. 1, No. 2, 251-260 (1993; Zbl 0787.06009)] introduced the notions of filter and implicative filter in a lattice implication algebra, and investigated their properties. In this paper we give an equivalent condition for a filter, and provide some equivalent conditions that a filter is an implicative filter in a lattice implication algebra. By using these results, we construct an extension property for implicative filters.


03G10 Logical aspects of lattices and related structures