### Abstract

Interval arithmetic is based on the fact that for intervals on the real line, the element-wise 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 element-wise 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 element-wise product and sum of two intervals are always intervals.

Original language | English |
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Pages (from-to) | 179-190 |

Number of pages | 12 |

Journal | Reliable Computing |

Volume | 4 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1998 |

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## Cite this

Vroegindeweij, P. G., & Kosheleva, O. M. (1998). When is the product of intervals also an interval?

*Reliable Computing*,*4*(2), 179-190. https://doi.org/10.1023/A:1009937210234