Interval Mathematics: Algebraic Aspects Sompong Dhompongsa

University of Texas at El Paso DigitalCommons@UTEP

Departmental Technical Reports (CS) Department of Computer Science

10-1-2001 Interval Mathematics: Algebraic Aspects Sompong Dhompongsa

Vladik Kreinovich University of Texas at El Paso, [email protected]

Hung T. Nguyen

Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: UTEP-CS-01-29. Published in the Proceedings of the Second International Conference on Intelligent Technologies InTech'2001, Bangkok, Thailand, November 27-29, 2001, pp. 30-38.

Recommended Citation Dhompongsa, Sompong; Kreinovich, Vladik; and Nguyen, Hung T., "Interval Mathematics: Algebraic Aspects" (2001). Departmental Technical Reports (CS). Paper 416. http://digitalcommons.utep.edu/cs_techrep/416

This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Interval Mathematics: Algebraic Aspects

Sompong Dhompongsa Vladik Kreinovich Hung T. Nguyen Chiang Mai University Dept. Computer Science Mathematics Dept. 239 Huay Kaew Road University of Texas New Mexico State University Chiang Mai 50200, Thailand El Paso, TX 79968, USA Las Cruces, NM 88003, USA [email protected] [email protected] [email protected] Abstract: Many application-oriented mathematical models deal with real numbers. In real life, due to the inevitable measurement inaccuracy, we do not know the exact values of the measured quantities, we know, at best, the intervals of possible values. It is thus desirable to analyze how the corresponding mathematical results will look if we replace numbers by intervals. Keywords: interval mathematics

1 Introduction: Why interval mathe- 2.2 Linear equations with interval coefﬁ-

matics? cients: polytope

A natural idea is to consider the matrix

! ¡ Mathematical models normally deal with real num- ¡

and the vector with interval coefﬁcients bers. In real life, due to the inevitable measurement ¡

and . Then, it is natural to deﬁne the solution

inaccuracy, we do not know the exact values of the !

# £

set as the set of all solutions of the system

measured quantities, we know, at best, the intervals ¡

£$ ¢ $ for all %$¨ and all , i.e., for which

of possible values. Thus, with applications in mind, ¡

$ & ' it is desirable to revisit computational formulas and and £ for all and . related mathematical results and see what they look

Deﬁnition 2.1. Let be an interval matrix, and let like if we use intervals instead of real numbers. (

be an interval vector. We say that a vector is a

¡ )

solution for the system of interval linear In particular, if we know only the intervals , for

equations if there exists a matrix %$¥ and a vector

¡ £

two numbers ¢ and , then the set of possible values

£$ for which . ¢¨§©£

of ¢¥¤¦£ (correspondingly, of ) also forms an

interval. It is natural to call this interval the sum ¤

¡ Theorem 2.1 [1]. For every system of interval linear

(corr., product) of the original intervals and . equations, its set of solutions is a polytope.

In other words, for every set of interval linear equa- 2 First algebraic aspect: Adding inter- tions, the border of its solution set is piece-wise lin- val data. How do intervals change ear (hyperplanar). simple algebraic shapes?

2.1 Simplest shape: (hyper)plane 2.3 Symmetric linear equations with interval coefﬁcients: piece-wise quadratic

The simplest possible shape is a hyperplane, which shape

* can be deﬁned as a solution set for a system of linear

When we consider symmetric interval matrices ,

equations i.e., matrices for which , then it is natural to

restrict ourselves only to symmetric matrices $+ .

§ £ ¢

Deﬁnition 2.2. Let be a symmetric interval ma-

(

£ ¢

i.e., in matrix form, , where , trix, and let be an interval vector. We say that a

£ £ ) , and . vector is a s-solution for the system of

interval linear equations if there exists a symmetric Deﬁnition 2.5. A set is called semialgebraic if it can

£$ # £ matrix $¥ and a vector for which . be represented as a ﬁnite union of subsets, each of

which is deﬁned by a ﬁnite system of polynomial

§§/§ ¦

*,+.- 102

Theorem 2.2 [2, 3, 6]. For every system of interval equations and inequalities of the

§/§/§ ¦ §§/§ ¦

*435- *879- 102"6 102;: linear equations , its set of s-solutions has a types and

piece-wise quadratic border. (for some polynomials * ).

In other words, the border of this set consists of

$>=

Deﬁnition 2.6. For every set < , and for ev-

§/§§ §/§§

ﬁnitely many pieces each of which can be described A

? & @0 #

ery subset & , by a

by a quadratic equation:

it projection B1CD-E

§/§§

£¢

¦¨§

- 2 $I= ¡

H that can be extended to a vector

§ § ¤ § ¤¥¤

§§/§

- 2 $J<

It is known that a projection of a semialgebraic set is 2.4 Linear equations with interval coefﬁ- also semialgebraic. Now, we are ready for the main

cients and general linear dependence results:

between © and : arbitrary algebraic

! * sets

! of a system, is also semialgebraic, and that, vice It

The symmetry relation ¢ is a particular case

is known that for every semialgebraic set < , and for

§§/§ §/§/§

of linear dependence between the coefﬁcients . A

& &@0 #

every subset ? , the cor-

The corresponding equations # can be de-

responding projection of < , i.e., the set of all vectors

¢ & ' §/§§

scribed if we ﬁx the values for , and then 0

- 2 $K=

H that can be extended to a solu-

¢ ¢ §/§/§

take .

2 tion - of a system, is also semialgebraic, Symmetry is just one of such relations. It is natural and that, vice to consider other relations such as antisymmetry, etc. In general, we arrive at the following definition: Theorem 2.3 [4]. For every system of interval linear equations with interval coefficients, the set of its Definition 2.3. By a system of interval linear equa- solutions is semialgebraic.

tions with dependent coefﬁcients, we mean a system

of the type Theorem 2.4 [4]. Every semialgebraic set can be

represented as a projection of the set of all solutions

£ of a system of interval linear equations with interval

coefﬁcients.

where

(

In this representation, we allow intervals to be ar-

¢ ¢ ¤ ¢

bitrarily wide. In terms of measurements, wide in-

tervals correspond to low measurement accuracy. It

is natural to ask the following question: if we only

£ £ ¤ £

consider narrow intervals, which correspond to high

measurement accuracy, will we still get all possible

¢ £ £

¢ , , , and are given real numbers, ( semialgebraic shapes or a narrower class of shapes?

! '"$# !&%'

& ), and coefﬁcients

¦ can take arbitrary values from the given intervals ( .

Deﬁnition 2.7 [10]. For a given LM6 , an interval

P P

(

;Q¥R ¤KRTS

is called absolutely L -narrow if P

Deﬁnition 2.4. We say that a vector is a solution

L RU L§DV ,V RU L , and is relatively -narrow if .

to a given system of linear interval equations with £

dependent coefﬁcients if it solves a system

L$6 )( for some values $ . Theorem 2.5 [5]. For every , every semialgebraic set can be represented as a projection of the To describe the set of such solutions, we must recall solution set of some system of interval linear equa- two notions: the notion of a semialgebraic set and tions with dependent coefﬁcients, whose intervals are

the notion of a projection: both absolutely and relatively L -narrow. 3 Second algebraic aspect: Adding in- To describe the class of all interval-rational func- terval operations Interval + rational = tions, we must recall the deﬁnition of an algebraic function: algebraic

Deﬁnition 3.2. An analytic function ¦ §/§/§

3.1 Rational functions

- 2 is called algebraic if

The set of rational functions can be deﬁned as the

§§/§ §§/§ ¦

smallest set of functions that contains constants and * - T- 2 2

variables and is closed under arithmetic operations

§/§/§

¤ Q §

* - 2

( ). for some polynomial for which

the partial derivative is not identically 0

( ). We say that the corresponding

§§/§

3.2 What happens if we add intervals: a

* - 2

polynomial deﬁnes the algebraic §§/§

question

T- 2 function .

A natural question is: what class of functions will be "

get if we add to this list the “interval” operation, i.e., Deﬁnition 3.3. Assume that is an open domain in

$#%"'& =

an operation that transforms a function of vari- = . We say that an algebraic function

§/§§

( (

ables and given intervals into the bounds can be locally represented by an interval-rational

§/§§

¦ ¨¦

2%$ for the range -

function if for almost every point

(

" there exists a neighborhood and an interval-

§§/§

( (

§§/§

- 2

rational function such that for all

§§/§

( - 2 - 2 - 2

§/§§ §/§§ §

from , .

( (

- 2 V $ $ Here, almost every is understood in the usual math- 3.3 Deﬁnitions and the main result ematical sense: a property is true for almost every point if the set of all points in which it is not true has

Deﬁnition 3.1. By interval-rational functions of Lebesgue measure 0.

§ § §

several real variables , we mean functions from the following class: Theorem 3.1 [8]. Every interval-rational function is

algebraic. ¡ Constants and variables themselves are Theorem 3.2 [8]. Every algebraic function can be

interval-rational functions. locally represented by an interval-rational function.

¡ If and ¢ are interval-rational functions, then

In other words, if we add an interval substitution op-

£¢ Q¤¢ §¥¢ ¢ ¤ , , , and are interval-rational eration to the original list of algebraic operations de- functions.

scribing rational functions, we get an arbitrary alge-

§§/§ §/§§

§¦ ¨¦ ©

If - is an interval- braic function.

§/§§

rational function, and are intervals,

then §§/§

3.4 A more computer-realistic notion of “al-

2 -

¢ most all” and the corresponding result

§/§§ §§/§

¨¦ §¦ ©

- In mathematics, “almost all” usually means “all

GDG

points, except for points from a set of Lebesgue

is an interval-rational function. measure 0” (or, “except for points from a set of a

§§/§ §/§§

small Lebesgue measure”). In the existing com-

§¦ ¨¦

2 If - is an interval-

§/§§ puters, however, only rational numbers are repre-

rational function, and are intervals,

sented. The set of all rational numbers is countable

then §§/§

and has, therefore, Lebesgue measure 0; so the stan-

- 2

¢ dard mathematical notion of “almost all” is not very

§§/§ §/§§

§¦ ¨¦

computer-realistic.

2 -

G G

In real life, when we say that “an algorithm is ap- §§/§

is an interval-rational function.

plied to real numbers ”, we usually mean

¡ Only functions that are obtained by these oper- that this algorithm is applied to rational numbers

§/§§ §§/§

¤ )1Q ) ations are interval-rational functions. ¤ that are close to , where

6 (

is the computer precision. So, if we ﬁx ) , there exists a neighborhood and an interval-

§§/§

# - 2

we can say that an algorithm works for real rational function such that for all

§/§§

( - 12 T- 12

numbers if it works ﬁne for all tuples from , .

§/§§

- ¤ )1Q 2

of rational numbers ¤ that are close to §/§/§

Theorem 3.3 [9]. Every interval-rational function is

- 2 . algebraic. Now, we have real-valued inputs on which the algorithm works ﬁne, and real-valued inputs on which Theorem 3.4 [9]. Every algebraic function can be it does not. For real-valued inputs, we can apply locally represented by an interval-rational function. Lebesgue measure. 4 Third algebraic aspect: algebraic

Deﬁnition 3.4. generalizations of intervals

¡ 4.1 Multi-D analogues of intervals: why?

)I6

Let be a real number. We say that points

§§/§ §/§§

What are the natural multi-D analogues of intervals?

¦ ¨¦ §¦

- 2 $'= - 2 are

One of the main goals of science is to predict what

Q V Q V ) &

) close if for all .

will happen in the world. In other words, to pre-

= &6 ) dict the values of different physical quantities. Tradi-

Let be a bounded set, and let

$ =

be a real number. We say that a point tional prediction algorithms take as input the current

§/§§

©¨ Q

) possibly belongs to if there exists a point values of some physical variables, and

1Q that is ) close to and that belongs to . use these value to predict the future state of the Uni-

verse. For example, in weather prediction, we take

§/§§

We say that a bounded set is

©¨

as input the values of the meteorological

¢¡

) 2 Q - small if the set of all points that

parameters in different points.

) possibly belong to has Lebesgue measure

¡ . There are two main ways to ﬁnd the values of :

by directly measuring them, and by using the abil- ¡ We say that a bounded set is small if for ev-

ity of an expert to give a reasonable estimate. Mea- ¡

ery , there exists a ) for which the set is surements are never 100% accurate. Expert estimate

¢¡

) 2 Q

- small. are also only approximate. As a result, we never

¤£

¡ know the exact values of the desired physical quan- =

We say that a set (not necessarily §/§§

tities . There are usually several different

R 6 bounded) is small if for every , the in-

¦¥ possibilities that are all consistent with all our mea-

R RTS tersection Q is small.

surements and expert estimates.

* - 2

We say that a property holds for P

For example, if the measured value of a tempera-

computer-realistically almost every if the set

ture is 35, and the accuracy of this measurement

V¢§4* - 12 *

of all for which is false is N

is , this means that the actual value of can be

¦ ¦

small.

equal to any number from the interval S .

- 2 If a property * is almost always true in this sense, For these different possibilities, the predicting algo- this means, crudely speaking, that for any given ¡ , rithm will lead to slightly different results. There-

we can choose a computer precision ) so that for all fore, instead of a single predicted result, we get the

except for a set of measure , we can guarantee set of possible future values.

- 2 that * is true even when we only know compo-

Each measurement bring an additional restriction on

)

nents with precision . §§/§

- 2 the set of all possible values of . We can now reformulate the above Deﬁnition 3.3 and As a result, the more measurements we take, the

Theorems 3.1 and 3.2 in terms of this new computer- more complicated the shape of this set can be. realistic deﬁnition of “almost all”. But we need to process this set in order to get the

set of possible future values, and processing odd- "

Deﬁnition 3.5. Assume that is an open domain shaped sets is computationally very complicated. So,

#%" & in = . We say that an algebraic function we need to approximate these sets by sets belonging

= can be computer-realistically locally represented to some pre-chosen family of simple sets (e.g., en-

by an interval-rational function if for computer- close by a ball, or by an ellipsoid, or by a paral-

§/§§

¦ §¦

2 $ " realistically almost every point - , lelepiped). What family should we choose?

4.2 The necessity to check consistency: mo- to sets that describe the uncertainty of our knowl-

¤+¢ © tivations edge, then a set © (that is obtained from by a

translation) is also a reasonable approximation. For N Measuring devices can go wrong; expert estimates

can be wrong. As a result, different conditions on example, assume that we are measuring time, and as

.¦ ¦

If we now change the starting point for measuring

start processing, it would be nice to check the ex-

time, e.g., take Q as the new starting point, then the

isting knowledge for possible inconsistencies. N

same result will be expressed as

© ¤ S In view of this necessity, it is desirable to choose the . This new set must also be a reason- family of approximating sets in such a way that this able approximation. check will not be computationally very complicated.

Deﬁnition 4.2. We say that a family ¥ of sets is

$ ¥

Let’s formulate this necessity in mathematical terms. translation-invariant if for every © , and for

J= © ¤ ¢ ¥

Assume that we have several pieces of knowledge every ¢+$ , the set also belongs to ; this

as a set of all the vectors that are consistent with Theorem 4.1 [11, 12]. Translates of a compact con- this particular knowledge. So, we instead of saying

vex set © allow checking consistency if and only if

that we have several pieces of knowledge, we can

§/§/§ ¨

¢¡ = say that we have several sets . Consistency means that it is possible that a certain

So, a convex compact set © can be a reasonable rep-

$£=

value satisﬁes all these properties, i.e., be-

§/§/§ resentation of uncertainty if and only if is a par-

¢¡

longs to all these sets (in other words,

¥£ ¥ ¥

§§/§ allelepiped. This result explains why parallelepipeds

¡ that ). are often used to describe uncertainty. How can we actually check consistency? Knowledge This result is based on the assumption that © is con- usually comes piece after piece, so a typical situation vex. However, we often have knowledge that does

is as follows. We already have a consistent knowl- not correspond to a convex set. E.g., if we have N

edge base. In other words, we have the pieces of ¦

measured the value of the velocity as (i.e.,

§§/§

knowledge represented by sets , and these the possible values form an interval /S ), and this pieces of knowledge are consistent. Then, a new is the only information that we have about the mo-

piece of knowledge arrives, described by a set . tion, then the set of all possible values of the velocity

components forms a (non-convex) “slice”

A natural way to check consistency is to check

between two spheres (of radii 9 and 11).

whether is consistent with each of the existing

sets . The consistency between and each of So, we arrive at the following open problem:

is, of course, necessary for the new knowledge base

§/§/§

Open problem. To describe sets for which translates

3 to be consistent. It is easy to see,

however, that this comparison is not always sufﬁcient allow checking consistency.

(the examples of why it is not always so can be easily

There exist non-convex compact sets with this prop-

£ ¦

extracted from the following text). So, we arrive at

= erty: e.g., translates of © allow the following deﬁnition: checking consistency. However, all such examples that we have constructed so far are disconnected. 4.3 Deﬁnitions, results, and hypotheses Therefore, we can formulate the following hypothesis:

Deﬁnition 4.1. We say that a family ¥ of sets allows checking consistency if the following is true: For ev-

§/§/§ Hypothesis 4.1. If translates of a compact connected

$§¥ $

ery ¦ , and for every tuple of sets

§/§§ set allow checking consistency, then is a par-

$¨¥

¥ , for which the sets are consis-

¥ ¥

§/§/§ allelepiped.

tent (i.e., ) and is consistent

§/§§

& ¦

with all (i.e., for all ), To help to solve our open problem, let’s reformulate

§§/§

¨¤ 3

all ¦ sets are consistent (i.e., it in terms close to those of a well-developed ﬁeld of

¥ ¥ ¥

§§/§

). math: namely, of homological algebra.

¨ =

We are interested in measurements, so it is natural Deﬁnition 4.4. Assume that a set © is ¦ to assume that if © is a reasonable approximation given, and a positive integer is ﬁxed. For every

# # Q

, cochains will be deﬁned as (completely) an- sum and the product of intervals is not always an in-

§§/§

tisymmetric functions from to a cer - terval. For example, on the ring of all integers, the

= L

tain subset © . A coboundary operator element-wise product

N N

Q - # ¤ 29Q

transforms an # cochain into an cochain

¢ ¢ S § ¢ ¢ S

N N

as follows:

§§/§

- L 29- & & 2

N N

¢ § ¢ FV ¢ $ ¢ ¢ S ¢ $ ¢ ¢ /S

§/§§ §/§/§

N N %

¢¡

¤£

-@Q 2 - & & & & 2

§5S § S §

of the intervals and

§5S©§ S §

is equal to ; this product

where means that we are skipping Q th variable. is not an integer-interval. When is it an interval?

§/§§

Comment. This definition is similar to standard def- Definition 4.5. A ring is a set of elements § initions from group cohomologies, with the only with two binary operations ¤ and (called addition difference that in our case, the range of the func- and multiplication) that satisfies the following three

tions deﬁned as cochains is not a group, it is a properties: ¨ subset of an additive group = . Because of that,

we have to be careful in our deﬁnition to guaran- ¡

is an Abelian (commutative) group under ad- Q tee that the coboundary of a # cochain will be an

dition;

#¥¤ 2 Q

- cochain. This is indeed guaranteed by our

erator: ¡ the right- and left-distributive laws hold, i.e.,

# Q

Theor em 4.2 [11, 12]. For every cochain ,

¢ §.- £ ¤ /2 ¢ § £ ¤ ¢ §

2 - L L .

and

# Q

Deﬁnition 4.5. An cochain is called an

- ¢ ¤ £92§¢ ¢ §¢ ¤ £ §

¢ Q L

# coboundary if it is a coboundary of some

- # QU 29Q ¢ # Q

cochain . An cochain is called an

Q L

# cocycle if . We say that a ring has no (proper) divisors of zero if

¦ ¦ ¦

§ £ ¢ £ from ¢ , it follows that or .

Comment. From Theorem 4.2, it follows that every

Q # Q

# coboundary is an cocycle. It turns out that the Deﬁnition 4.6. By an ordered ring, we mean a ring inverse statement is true if and only if translates of with a partial order such that:

¢;6

there exists an element ¢ for which ; =

lates of allow checking consistency if and only if ¡

£ $ ¢ ¤

if ¢ , then for every , we have

Q § Q

every § cocycle is a coboundary.

£ ;

¦ ¦ ¦ ¡

Comment. In traditional homology theory, both

6 £ 6 ¢ § £ 6

if ¢ and , then ;

# Q ¨

the set of all coc ycles and the set of all

# Q

coboundaries are abelian groups. Since

¨ , in this traditional theory, the condition An order is called consistent if the following two

that every 2-cocycle is a 2-cobounbdary can be re- properties hold:

¦ ¦ ¦ ¨

is called the second cohomology group. ¡

6 ¢ § £ 6 £ 6

if ¢ and then ;

¦ ¦ ¦

6 ¢ §£ 6 ¢;6 4.4 Interval operations in an arbitrary ring if £ and then . An even more general idea is to consider intervals in

an arbitrary partially ordered ring. For such rings, An order is called linear or total if for every two ele-

£ ¢ £ ¢ £ ¢;6 £ unlike the ring of real numbers, the element-wise ments ¢ and , either , or , or . Comment. In the following text, we will only con- or non-commutative). In some algebra textbooks, a sider consistently ordered rings with no divisors of ﬁeld is deﬁned as a commutative ring with a property

zero. that non-zero elements form a group. In such books, N

a non-commutative ring with this property is called a

Deﬁnition 4.7. Let be an ordered ring with no

skew ﬁeld, or a s-ﬁeld.

¢ S

divisors of 0. By an interval, we mean a set ¢

UV ¢ ¢) ¢ ¢ ¢ ¢ ¢¨$ , where . The element Not every ring is invertible (for example, as we have is called the lower endpoint of this interval, and the already mentioned, the ring of all integers is not in-

element ¢ is called its upper endpoint. The sum and vertible). There can be two reasons for that: product of two intervals are deﬁned element-wise.

¡ In some cases, multiplication is not invertible in We want to reformulate the desired property — that the original ring, but we can still easily add in- the element-wise sum and element-wise product of termediate elements and make it invertible. For two intervals are always intervals — in purely alge-

example, in the ring of integers, we do not have

¢ ¢

braic terms. §

a number for which § , but we have

§ § § § 6

§ and , so, we can add an extra

This property is true for the ring of real numbers and

§ § it is not true for the ring of integers. From the alge- element 3/2 for which . If we add braic viewpoint, the main difference between these all such elements, we will thus expand the ring

two rings is as follows: of integers to the ﬁeld of all rational numbers. ¡

It can also be that an element is “inﬁnitely ¡

smaller” than ¢ in the sense that whatever ele- ¢

in the ring of real numbers, multiplication is ¢

invertible in the sense that for every two real ment we take, we always have )§ .

numbers and , there exists a number

¢ ¢

§ ¢

( ¢ ) for which ; Rings in which this second reason is the only obsta-

¡ cle to invertibility will be called almost invertible:

on the other hand, multiplication in the ring of

all integers is not invertible: for example, § ,

¢ ¢

Deﬁnition 4.9. Let be an ordered ring.

but there exists no integer for which § .

We say that an element is left-inﬁnitely

In view of this difference, it is natural to conjecture smaller than an element ¢ (and denote it by

¦ ¦

¤£ ¢ § ¢ $ that the desired interval property is equivalent to in- ), if for all .

vertibility of multiplication. This ﬁrst guess turns

¡ out to be wrong: there are examples of ordered rings We say that an element is right-inﬁnitely

with the above interval property in which multiplica- smaller than an element ¢ (and denote it by

¦ ¦

¤£ ¢ § ¢ $ tion is not invertible. These example enable to cor- ), if for all . rect the original conjecture into an exact theorem, ac-

cording to which the interval property is equivalent Deﬁnition 4.10. Let be an ordered ring. We say

to a special property called almost invertibility. that multiplication is almost invertible (and, respec-

tively, that the ring is almost invertible) if for ev-

To introduce this new notion, let us ﬁrst recall the ¦

ery ¢ and , the following two properties hold:

formal deﬁnition of invertibility:

if is not left-inﬁnitely smaller than , then ¢

Deﬁnition 4.8. Let be a ring. We say that multi- ¢

§ ¢

plication is invertible if for every two elements ¢ and there exists an element for which .

, there exist: ¡ ¢

if is not right-inﬁnitely smaller than , then

¢ ¢

§ ¢

there exists an element for which .

¢ ¢

§ ¢

an element for which ; and

¢ ¢

¡ Now, we are ready to formulate the main result of

+ +

¢ § an element for which .

this section:

If a ring has a unit element ¡ (i.e., with an element Theorem 4.4 [7]. For every consistently ordered

§ ¢ ¢)§¢¡ ¢ ¢ for which ¡ for all ), then a ring ring with no divisors of 0, the following two con- is invertible if and only if it is a ﬁeld (commutative ditions are equivalent to each other: i) The product of every two intervals is also an in- 4.6 Auxiliary results terval. We have found the conditions under which both the product and the sum of the two intervals form an

ii) The ring is totally (linearly) ordered and al- interval. It may be interesting to consider the case most invertible. when only addition is defined, and find out when, in this case, the sum of any two intervals is also an in- If one of these conditions is satisfied, then the sum of terval. every two intervals is also an interval. Definition 4.12. By an ordered (Abelian) group, we For the rings in which there are no infinitely smaller mean an Abelian group with a partial order such elements, this result can be further simplified: that:

Deﬁnition 4.11. We say that a ring is weakly ¦

¢;6 there exists an element ¢ for which ;

Archimedean if none of its elements is left- or right-

£ $ ¢ ¤

inﬁnitely smaller than any other. if ¢ , then for every , we have

¤ £ . Many rings are weakly Archimedean, including the ring of all real numbers, the ring of all integers, and To formulate our result, we will need the following many others. A weakly Archimedean ring is almost property: invertible if and only if it is invertible. Hence, we get

the following corollary: Deﬁnition 4.13. We say that an order is 2-2-

£ ¢ £ separating if for every four elements ¢ , if each

Corollary 4.1 [7]. For every weakly Archimedean £ of the two elements ¢ and is smaller than or equal to

consistently ordered ring with no divisors of 0,

£ ¢ ¢ ¢ £ £ ¢ each of the elements ¢ , (i.e., if , , ,

the following two conditions are equivalent to each

£ and £ ), then there exist an element that lies in

other:

¢ £ £ between, i.e., for which ¢ and .

i) The product of every two intervals is also an in- Comments. terval. ¡ This deﬁnition is close to the notion of TR(2,2)

ii) The ring is linearly ordered and invertible. (Riesz) ordered groups.

If one of these conditions is satisﬁed, then the sum of ¡ Every group in which an order forms a lattice

every two intervals is also an interval. has this property.

We have already mentioned that a ring with a unit ¡ If a group is a compact topological group in element is invertible if and only if it is a ﬁeld. Hence, which the order is (in some reasonable sense) we get the following second corollary: consistent with topology, then this property is equivalent to being a lattice.

Corollary 4.2 [7]. For every weakly Archimedean

¡ consistently ordered ring with a unit element and Theorem 4.5 [7]. Let be an ordered Abelian with no divisors of 0, the following two conditions group. Then, the sum of every two intervals is also are equivalent to each other: an interval if and only if the order on the group is 2-2-separating. i) The product of every two intervals is also an interval. Acknowledgments This work was supported in part by NASA under co- ii) The ordered ring is a linearly ordered field. operative agreement NCC5-209 and grant NCC 2- If one of these conditions is satisfied, then the sum of 1232, by the Future Aerospace Science and Tech- every two intervals is also an interval. nology Program (FAST) Center for Structural In- tegrity of Aerospace Systems, effort sponsored by In short, this result says that for interval arithmetic to the Air Force Office of Scientific Research, Air Force be possible in a ring, this ring has to be a field. It is Materiel Command, USAF, under grant number worth mentioning that, somewhat in contrast to this F49620-00-1-0365, by NSF grants CDA-9522207 result, the intervals themselves do not form a field; and 9710940 Mexico/Conacyt, and by Grant No. W- for example, the set of intervals does not even have a 00016 from the U.S.-Czech Science and Technology distributivity property. Joint Fund. References [11] D. Misane and V. Kreinovich, “A new charac- terization of the set of all intervals, based on the [1] W. Oettli and W. Prager, “Compatibility of Ap- necessity to check consistency easily”, Reliable proximate Solution of Linear Equations with Computing, 1995, Vol. 1, No. 3, pp. 285–298. Given Error Bounds for Coefficients and Right– hand Sides”, Numer. Math., 1964, Vol. 6, pp. [12] D. Misane and V. Kreinovich, “The necessity 405–409. to check consistency explains the use of parallelepipeds in describing uncertainty”, Geombi- [2] G. Alefeld, V. Kreinovich, and G. Mayer, “Sym- natorics, 1996, Vol. 5, No. 3, pp. 109–120. metric Linear Systems with Perturbed Input Data”, In: G. Alefeld and J. Herzberger (eds.), Numerical Methods and Error Bounds. Pro- ceedings of the IMACS-GAMM International Symposium on Numerical Methods and Error Bounds, Oldenburg, Germany, July 9–12, 1995, Akademie Verlag, Berlin, 1996, pp. 16–22. [3] G. Alefeld, V. Kreinovich, and G. Mayer, “The Shape of the Symmetric, Persymmetric, and Skew–Symmetric Solution Set”, SIAM Journal on Matrix Analysis and Applications (SIMAX), 1997, Vol. 18, No. 3, pp. 693–705. [4] G. Alefeld, V. Kreinovich, and G. Mayer, “The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients”, Mathematische Nachrichten, 1998, Vol. 192, pp. 23–36. [5] G. Alefeld, V. Kreinovich, G. Mayer, and M. Huth, “A Comment on Shape of the Solu- tion Set for Systems of Interval Linear Equations with Dependent Coefficients”, Reliable Comput- ing, 2001, Vol. 7, No. 3, pp. 275–277. [6] G. Alefeld, G. Mayer, and V. Kreinovich, “The shape of the symmetric solution set”, In: R. B. Kearfott et al (eds.), Applications of Inter- val Computations, Kluwer, Dordrecht, 1996, pp. 61–79. [7] O. Kosheleva and P. G. Vroegindeweij, “When is the product of intervals also an interval?”, Re- liable Computing, 1998, Vol. 4, No. 2, pp. 179– 190. [8] V. Kreinovich, “Interval rational = algebraic”, ACM SIGNUM Newsletter, 1995, Vol. 30, No. 4, pp. 2–13. [9] V. Kreinovich and A. V. Lakeyev, “ ‘Interval Ra- tional = Algebraic’ Revisited: A More Com- puter Realistic Result”, ACM SIGNUM Newslet- ter, 1996, Vol. 31, No. 1, pp. 14–17. [10] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibil- ity of data processing and interval computations, Kluwer, Dordrecht, 1997.