Hilbert Implication Algebra and Some Properties

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.

The logic induced by effect algebras

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras $${\mathbf {E}}$$ E , we investigate a natural implication and prove that the implication reduct of $${\mathbf {E}}$$ E is term equivalent to $${\mathbf {E}}$$ E . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

Lattice of closure endomorphisms of a Hilbert algebra

A closure endomorphism of a Hilbert algebra [Formula: see text] is a mapping that is simultaneously an endomorphism of and a closure operator on [Formula: see text]. It is known that the set [Formula: see text] of all closure endomorphisms of [Formula: see text] is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of [Formula: see text], anti-isomorphic to the lattice of certain closure retracts of [Formula: see text], and compactly generated. The set of compact elements of [Formula: see text] coincides with the adjoint semilattice of [Formula: see text]; conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.

Absorbent Li – Ideals of Lattice Implication Algebras

on this paper we present the concept of Absorbent LI – ideal of pass section notion polynomial math L. We communicate about the relations some of the Absorbent LI – requirements to ILI – beliefs, Associative LI – goals and awesome LI – desires of L. We exhibit the augmentation hypothesis of Absorbent LI – necessities and the crossing factor of Absorbent LI – dreams is additionally an Absorbent LI – best. We exhibit that the complement of Absorbent LI – first-rate is Absorbent channel. At lengthy closing we observe dubious idea to Absorbent LI – requirements and talk approximately the superb houses of hard to recognize Absorbent LI – desires.

Q-Filters of Quantum B-Algebras and Basic Implication Algebras

The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established.