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On the Axiomatization of Interval Arithmetic

We investigate some abstract algebraic properties of the system of intervals with respect to the arithmetic operations and the relation inclusion and derive certain practical consequences from these properties. In particular, we discuss the use of improper intervals (in addition to proper ones) and of midpoint-radius presentation of intervals. This work is a theoretical introduction to interval arithmetic involving improper intervals. We especially stress on the existence of special “quasi”-multiplications in interval arithmetic and their role in relevant symbolic computations.