Kites and residuated lattices

We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.     

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.     

Fuzzy quantum spaces and compatibility

Using ideas from fuzzy set theory, we generalize the notion of a quantum (probability) space introduced by Suppes to so-called fuzzy quantum spaces and we study the compatibility problem.     

Generalization of Maeda's theorem

The theorem of S. Maeda concerning the characterization of finite measures on a quantum logic of all closed subspaces of a Hilbert space of dimension 2 is generalized to the case of-finite measures with possible infinite values. The proof does not involve Gleason's result, but only the proposition on frame functions.     

Gleason's theorem and Cauchy's functional equation

We study measures on the effect algebras of the closed interval [0, 1] and we describe regular or bounded measures. Applying Gleason's theorem for measures on the system of all closed subspaces of a Hilbert space, we show that any bounded measure m has the formm(t)=tm 1, t [0, 1], and as a by-product it gives a solution of Cauchy's basic functional equationf (x+ y)=f(x) + f(y) forx, y,x + y [0, 1].     

Product of partition logics,orthoalgebras, and automata

We attempt to define a coupled system consisting of two partition logics and we introduce a product of partition logics. These partition logics have a close connection with Moore and Mealy-type automata. We show how the coupled system of two automata is connected with the product of partition logics, and present some illustrative examples.     

On Gleason’s Theorem without Gleason

The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In particular, if dim H=n, then on ℒ(H) there is a unique discrete state, namely m(A)=dim A/dim H, A∈ℒ(H). Dedicated to Pekka J. Lahti on the occasion of his 60th birthday. The paper has been supported by the Center of Excellence SAS–Physics of Information–I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contract APVV-0071-06, Bratislava, Slovakia.     

Probabilistic Averaging in Bounded Rℓ-Monoids

Bounded Rℓ-monoids generalize GMV-algebras and pseudo BL-algebras. Such monoids do not admit, in general, any analogue of addition, in contrast to GMV-algebras. Nevertheless we introduce the notion of a state (an analogue of a probability measure). It coincides with that for GMV-algebras. We show that the existence of states is crucially connected with the existence of normal and maximal filters. In addition, some topological properties of the extremal states and the hull-kernel topology of filters are studied.