When is the product of intervals also an interval? Olga Kosheleva1 and Piet G. Vroegindeweij2 1Department of Electrical and Computer Engineering University of Texas at El Paso El Paso, TX 79968, USA email [email protected] 2Koenestraat 10 NL 3958 XG Amerongen, the Netherlands 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, multi- plication, 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. 1 Introduction Intervals are important. Measurements are never 100% precise. As a result, if the measurement result is, say, 100, this does not mean that the actual value is exactly 100, it rather means that the actual value must be within an interval [100 ¡ ¢; 100 + ¢], where ¢ is an upper bound on the measurement error (this upper bound is usually supplied by the manufacturer of the measuring instrument). For example, if ¢ = 1, then the actual value can be equal to any number within the interval [99; 101]. Similarly, if we, say, measure the total charge to determine the total amount of electrons in a given beam, and the measurements are not accurate enough, we do not get the exact number of electrons, but only the interval of possible values, such as [99; 101], meaning that there can be either 99, or 100, or 101 electrons (i.e., that [99; 101] = f99; 100; 101g). 1 In both cases, after the measurement, we get not the exact value of the measured quantity, but only the interval of its possible values: a = [a; a] = fa 2 X j a · a · ag: (1) In the ¯rst example, the set X of possible actual values of the measured quantity is a set R of all real numbers; in the second example, X coincides with the set of all integers. Intervals on the real line: Traditional interval arithmetic. Often, we need to process the measured data. If the measured quantities are real numbers, then the measurement results can be expressed by intervals on the real line. To process such \real-interval" data, we can use interval arithmetic originally proposed by R. E. Moore (for recent surveys, see, e.g., [2, 3]). In general, in data processing, we are interested in the quantity b = f(a1; : : : ; an) that is related, in a known way, with directly measured quantities a1; : : : ; an. To describe possible values of a, we measure the quantities a1; : : : ; an, and use the resulting intervals a1 = [a1; a1];:::; an = [an; an] of possible values of ai to describe the set B of possible values of b as B = fb j b = f(a1; : : : ; an) for some a1 2 [a1; a1]; : : : ; an 2 [an; an]g: (2) The resulting set B is called the range of the function f(a1; : : : ; an) on the intervals a1;:::; an, or the result of applying the function f(a1; : : : ; an) to the intervals a1;:::; an. In particular, if the data processing operation f(a1; : : : ; an) consists of simply applying an arithmetic operation + or ¤, i.e., if f(a1; a2) = a1 + a2 or f(a1 ¤ a2) = a1 ¤ a2, then we get the element-wise de¯nitions of the sum and of the product of two intervals: [a1; a1]+[a2; a2] = fb j b = a1 +a2 for some a1 2 [a1; a1] and a2 2 [a2; a2]g; (3) [a1; a1] ¤ [a2; a2] = fb j b = a1 ¤ a2 for some a1 2 [a1; a1] and a2 2 [a2; a2]g: (4) For intervals on the real line, thus de¯ned element-wise sum and product are also intervals (and the formulas for computing the endpoints of these intervals form the basis of interval arithmetic). Intervals on the set of integers are more di±cult to handle than in- tervals on the real line. For intervals on the set of all integers, we can also de¯ne element-wise addition and multiplication by using formulas (3) and (4), but these integer-intervals are more computationally di±cult to handle because the product of two integer-intervals is not always an interval: e.g., the element- wise product of two integer-intervals [1; 2] = f1; 2g and [1; 3] = f1; 2; 3g is equal to [1; 2] ¤ [1; 3] = f1; 2; 3; 4; 6g; this product is not an integer-interval. 2 A natural question. A natural question is: How can we describe all algebraic structures for which the product of two intervals is always an interval? To formulate this problem in precise terms, we need to have an algebraic struc- ture with addition, multiplication, and order that are (in some reasonable way) consistent. In the next section, we show that natural consistency demands lead to a notion of a consistently ordered ring. Thus, in algebraic terms, we get the following problem: Given an arbitrary consistently ordered ring K, we can de¯ne another structure of interval objects of the form fa 2 K j a · a · ag, with addi- tion and multiplication de¯ned element-wise. We want to describe all the rings for which this new interval structure is closed under addition and multiplication. 2 Motivation for the Following De¯nitions We need two consistent operations and one relation. We want to for- mulate our problem in general algebraic terms. Since our question is about addition and multiplication of intervals, this structure must contain addition, multiplication, and the order (so that we will be able to de¯ne intervals); we must also require that these operations be consistent with each other. Addition. First, we need an operation called addition. We will assume that addition + is commutative and associative, that there exists an element 0 for which 0 + a = a for all a, and that for every element a, there is an inverse element ¡a for which a + (¡a) = 0. In other words, we will assume that our structure K is an Abelian (commutative) group under addition. Multiplication. Second, we need an operation called multiplication. It is natural to require that multiplication is associative and distributive with respect to addition (i.e., (a + b) ¤ c = a ¤ c + b ¤ c and a ¤ (b + c) = a ¤ c + b ¤ c). These properties de¯ne a ring. So, we will consider rings. Comment. Multiplication of real numbers has an additional property: it is com- mutative (a¤b = b¤a). However, we do not want to impose this restriction on the general structure, because there are meaningful computation-related examples of non-commutative ordered rings: e.g., a ring of all n £ n real-valued square matrices over A (with A ¸ 0 i® A is non-negative de¯nite) is non-commutative. Order. In order to de¯ne intervals on a ring, we must have an order. The above matrix example shows that the order need not be total (linear); therefore, in this paper, we will consider the general case of a (partial) order. This order 3 must be consistent with + and ¤. There are several reasonable requirements that express this consistency: ² First, there must exist a positive element a > 0. ² Second, the order must be shift-invariant: if a < b, then for every c, we must have a + c < b + c. ² Third, the sum and the product of two positive numbers must be positive. ² Fourth, if a ¤ b is positive and a is positive, then b must be positive too, and if both a ¤ b and b are positive, then a must be positive. There is a ¯fth natural property that does not need to be speci¯cally postulated for linear orders, but which is needed for partial orders: the product of two non- zero numbers must be non-zero. (If we have a linear order, then every non-zero number is either positive or negative, and this property simply follows from the third one.) These ¯ve properties are true for all three examples that we had considered so far: for the ring of all real numbers, for the ring of all integers (with natural order in both examples), and for the ring of all n £ n square matrices. The notions described by these properties are well known in algebra: an object that satis¯es the ¯rst three properties is called an ordered ring; an ordered ring that satis¯es the fourth property as well is called a consistently ordered ring, and a ring that satis¯es the additional ¯fth property is called a ring with no divisors of zero. In these terms, in the following text, we will consider consistently ordered rings with no divisors of 0. We are now ready to introduce formal de¯nitions: 3 De¯nitions and the Main Result 3.1 De¯nitions of the algebraic structure: consistently or- dered rings, intervals, operations with intervals De¯nition 1. (see, e.g., [4]) ² A ring K is a set of elements with two binary operations + and ¤ (called addition and multiplication) that satis¯es the following three properties: { K is an Abelian (commutative) group under addition; { multiplication is associative; { the right- and left-distributive laws hold, i.e., a ¤ (b + c) = a ¤ b + a ¤ c and (a + b) ¤ c = a ¤ c + b ¤ c.