Max-min representation of piecewise linear functions.
Lattical token systems.
Stochastic token theory is a new branch of mathematical psychology. In this paper we investigate algebraic properties of token systems defined on finite lattices.
A stochastic model of choice.
An approach to choice function theory is suggested which is probabilistic and non-deterministic. In the framework of this approach fuzzy choice functions are introduced and a number of necessary and sufficient conditions for a fuzzy choice function to be a fuzzy rational choice function of a certain type are established.
Involutions in fuzzy set theory.
All possible involutions in fuzzy set theory are completely described. Any involution is a composition of a symmetry on a universe of fuzzy sets and an involution on a truth set.
Convexity in subsets of lattices.
The notion of convex set for subsets of lattices in one particular case was introduced in [1], where it was used to study Paretto's principle in the theory of group choice. This notion is based on a betweenness relation due to Glivenko [2]. Betweenness is used very widely in lattice theory as basis for lattice geometry (see [3], and, especially [4 part 1]). In the present paper the relative notions of convexity are considered for subsets of an arbitrary lattice. In section...
Topological automorphism groups of chains.
It is shown that any set-open topology on the automorphism group A(X) of a chain X coincides with the pointwise topology and that A(X) is a topological group with respect to this topology. Topological properties of connectedness and compactness in A(X) are investigated. In particular, it is shown that the automorphism group of a doubly homogeneous chain is generated by any neighborhood of the identity element.
On fuzzy binary relations.
A binary relation language is an important tool of the theory of measurements (see, for example, book [5]). Specifically, the theory of nominal and ordinal scales is based on theories of equivalent relations and weak orderings. These binary relations have a simple structure which can be described as follows (bearing in mind a context of the measurement theory).
Extensions of set functions.
We establish a necessary and sufficient condition for a function defined on a subset of an algebra of sets to be extendable to a positive additive function on the algebra. It is algo shown that this condition is necessary and sufficient for a regular function defined on a regular subset of the Borel algebra of subsets of a given compact Hausdorff space to be extendable to a measure.
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