Implicative semi-lattices

Abstract

WILLIAM C. NEMITZ(') 1. Introduction. An implicative semi-lattice is an algebraic system having as models logical systems equipped with implication and conjunction, but not possessing a disjunction. The position of implicative semi-lattices in algebraic logic is clearly displayed in [2]. In [1], the relation of implicative lattices to Brouwerian logics is explained. In the terminology of [1], an implicative lattice would be called a relatively pseudo-complemented lattice. Monteiro [6] and Glivenko [5] have each given a set of axioms for Brouwerian logic. Glivenko has also proved certain results for Brouwerian logics (equipped with disjunction) which are proved for implicative semi-lattices in §4 of this paper. In [3] and [7], the relation of implicative lattices to closure algebras and topologies is explored. There are four principal results in this paper. The first (Theorem 3.2) states that the relationship between homomorphisms of implicative semi-lattices and their kernels is that usually found in algebra, and not found in lattice theory, namely, that every homomorphism is essentially determined by its kernel. In [4], Frink has shown that the structure of a pseudo-complemented semi-lattice, and therefore of a bounded implicative semi-lattice, is very similar to that of a bounded implicative lattice, in that the semi-lattice of "closed" elements is a Boolean algebra, despite the absence of a least upper bound. The second result of this paper (Theorem 4.4) states that every element of a bounded implicative semi-lattice is the meet of its closure with a dense element, a result which is well known for bounded implicative lattices. The third result (Theorem 5.2) gives a method for constructing all bounded implicative semi-lattices having a given Boolean algebra for closed algebra, and a given implicative semi-lattice for dense filter; the fourth result (Theorem 6.6) gives essentially the same information about homomorphisms.

Interval valued $$(\in,\in\!\vee\,q)$$ -fuzzy filters of pseudo BL-algebras

“…Further, Davvaz and Corsini (2007) redefined fuzzy H v -submodule and many valued implications. We also know that the implication operators were mentioned by Curry (1965) and also studied by Nemitz (1965) for semilattices. In Zhan et al (2008) also discussed the properties of interval valued (∈, ∈ ∨ q)-fuzzy hyperideals in hypernear-rings.…”

“…Using prime filters of B L-algebras, he proved the completeness of Basic Logic B L. B L-algebras are further discussed by Di Nola et al (2000), Di Nola and Leustean (2003), Iorgulescu (2004), Ma et al (2007) and Turunen (2001), Turunen and Sessa (2001), and so on. Recent investigations are concerned with non-commutative generalizations for these structures (see Deschrijver 2007, Di Nola and Leustean 2003, Dvurečenskij 2001a,b, 2007, Flonder et al 2001, Ma et al 2007, Nemitz 1965, Pu and Liu 1980, Rachunek 2002a, Zadeh 2005, Zeng and Li 2006. In Georgescu and Iorgulescu (2001), introduced the concept of pseudo M V -algebras as a non-commutative generalization of M V -algebras.…”

On Negatively Ordered Implicative Semigroups

“…Theorem 2.5. Let e; f be idempotents of a negatively partially ordered semigroup S; ; 6. Then e 6 f in S if and only if e = ef = f ein ES.…”

Ordered semigroups characterized by interval valued (ε, ε $\vee$ q)-fuzzy bi-ideals

“…This kind of semigroups has recently attracted a number of authors because such kind of semigroups provides the mathematical and logical foundation for informatic science and computer technology. Implicative semigroups are generalization of implicative semilattice (see Nemitz [9]) and has a close relation with implication in mathematical logic and set theoretic difference (see Birkhoff [1]). In [8], Ma et al considered the application of lattice-valued logic based on lattice implication algebras in machine intelligence.…”