Characterization Of Boolean Algebras Research Articles

Characterization of Boolean Algebras in Terms of Certain States of Jauch-Piron Type

Suppose that L is an orthomodular lattice (a quantum logic). We show that L is Boolean exactly if L possesses a strongly unital set of weakly Jauch-Piron states, or if L possesses a unital set of weakly positive states. We also discuss some general properties of Jauch-Piron-like states.

Automata theory based on quantum logic: some characterizations

Automata theory based on quantum logic (abbr. l -valued automata theory) may be viewed as a logical approach of quantum computation. In this paper, we characterize some fundamental properties of l -valued automata theory, and discover that some properties of the truth-value lattices of the underlying logic are equivalent to certain properties of automata. More specifically (i) the transition relations of l -valued automata are extended to describe the transitions enabled by strings of input symbols, and particularly, these extensions depend on the distributivity of the truth-value lattices (Proposition 3.1); (ii) some properties of the l -valued successor and source operators and l -valued subautomata are demonstrated to be equivalent to a property of the truth-value lattices which is exactly equivalent to the distributive law (Proposition 4.3 and Corollary 4.4). This is a new characterization of Boolean algebras in the framework of l -valued automata theory; (iii) we verify that the intersection of two l -valued subautomata is still an l -valued subautomaton if and only if the multiplication (&) is distributive over the union in the truth-value lattices (Proposition 4.5), which is strictly weaker than the usual distributivity; (iv) we show that some topological characterizations in terms of the l -valued successor and source operators also rely on the distributivity of truth-value lattices (Theorem 5.6). Finally, we address some related topics for further study.

On Kelley’s intersection numbers

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley’s characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number.

A combinatorial characterization of Boolean algebras

In this paper we prove that a groupoid is term equivalent to a Boolean algebra if and only if the number of n-ary term operations of the groupoid is equal to \(2^{2^n}\) for n = 0, 1, 2 and 3. This yields a partial solution of a problem posed by Berman in 1986.

Bell-type inequalities in orthomodular lattices. I. Inequalities of order 2

We study Bell-type inequalities of ordern with emphasis on the casen = 2 in the framework of the structure of an orthomodular lattice, which is a logicoalgebraic model of quantum mechanics. We give necessary and sufficient conditions for the validity of Bell-type inequalities of order 2. In particular, we study Bell-type inequalities in various structures connected with a Hilert space, and we give a characterization of Boolean algebras via the validity of certain Bell-type inequalities.

Common extension of Boolean informational systems

The notion of numerical characterization of Boolean algebras and coproducts are used to define information systems and to develop the theory of such systems.

Stone Lattices. I: Construction Theorems

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

A Characterization of Boolean Algebras

A Characterization of Boolean Algebras