In this paper we will study functions G of two variables on a quantum logic L, such that for each compatible elements a; b 2 L; Gða; bÞ ¼ mða^bÞ or Gða; bÞ ¼ mða _ bÞ or Gða; bÞ ¼ mðaMbÞ; where m is a state on L.

This paper treats the problem of fitting general aggregation operators with unfixed number of arguments to empirical data. We discuss methods applicable to associative operators (t-norms, t-conorms, uninorms and nullnorms), means and Choquet integral based operators with respect to a universal fuzzy measure. Special attention is paid to k-order additive symmetric fuzzy measures.

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown, that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our investigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.

In this paper we study conditions for the existence of a 3-dimensional s-map on a quantum logic under assumption that marginal s-maps are known. We show that the existence of such a 3-dimensional s-map depends on the triangle inequality of d-map, which on a Boolean algebra represents a measure of symmetric difference.

No abstract