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On the Algebra of Intervals and Convex Bodies
 J. UCS
, 1998
"... Abstract: We introduce and study abstract algebraic systems generalizing the arithmetic systems of intervals and convex bodies involving Minkowski operations such as quasimodules and quasilinear systems. Embedding theorems are proved and computational rules for algebraic transformations are given. ..."
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Abstract: We introduce and study abstract algebraic systems generalizing the arithmetic systems of intervals and convex bodies involving Minkowski operations such as quasimodules and quasilinear systems. Embedding theorems are proved and computational rules for algebraic transformations are given.
When is the product of intervals also an interval
 Reliable Computing
, 1998
"... Interval arithmetic is based on the fact that for intervals on the real line, the elementwise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is ..."
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Interval arithmetic is based on the fact that for intervals on the real line, the elementwise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is an elementwise product or a sum of two intervals always an interval? In this paper, we analyze this problem in a general algebraic setting: we need the corresponding algebraic structures to have (related) addition, multiplication, and order; thus, we consider (consistently) ordered rings. We describe all consistently ordered rings for which the elementwise product and sum of two intervals are always intervals. 1
In particular we have
, 809
"... Abstract. In this paper we present the set of intervals as a normed vector space. We define also a fourdimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some cases an euclidian division. 1. Interva ..."
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Abstract. In this paper we present the set of intervals as a normed vector space. We define also a fourdimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some cases an euclidian division. 1. Intervals and generalized intervals An interval is a connected closed subset of R. The classical arithmetic operations on intervals are defined such that the result of the corresponding operation on elements belonging to operand intervals belongs to the resulting interval. That is, if ⋄ denotes one of the classical operation +, −, ∗, we have (1)