On two interval-arithmetic structures and their properties

Abstract

Two interval-arithmetic structures are considered, which are extensions of the interval arithmetic for normal intervals using only familiar (outer) operations in two directions: i) via additional introduction of special inner (non-standart) operations; ii) vie extension of the set of normal intervals up to the set of directed (generalized) intervals by improper intervals. New propertiesof the interval structures are considered. The notations are unified by introducing new $(\pm )$-type indeces for the interval variables for an easy comparison and systematization of both structures. The so-called normal form for the representation of directed intervals is introduces, which allows to reformulate propositions for directed intervals into corresponding propositions for normal intervals, using inner and outer operations. Some propositions supporting the computation of functional ranges via interval arithmetic expressions are formulated.