Reliable Computing 7: 113–127, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
On the Algebraic Properties of Intervals and
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,
Block 8, BG-1113 Soﬁa, Bulgaria, e-mail: email@example.com
(Received: 3 May 1999; accepted: 10 October 1999)
Abstract. The algebraic properties of interval vectors (boxes) are studied. Quasilinear spaces with
group structure are studied. Some fundamental algebraic properties are developed, especially in
relation to the quasidistributive law, leading to a generalization of the familiar theory of linear spaces.
In particular, linear dependence and basis are deﬁned. It is proved that a quasilinear space with group
structure is a direct sum of a linear and a symmetric space. A detailed characterization of symmetric
quasilinear spaces with group structure is found.
Many-dimensional intervals (interval vectors, boxes) have been increasingly used
in interval analysis and reliable computing. Therefore it is necessary to study their
algebraic properties and the operations and relations between them. Interval vectors
form a quasilinear space with respect to addition and multiplication by scalar in the
sense of . A quasilinear space is an abelian cancellative monoid with respect
to addition, in particular, the monoid can be a group. By means of the familiar
extension method (used, e. g., when deﬁning negative numbers) any quasilinear
space can be embedded into a quasilinear space which is a group . Therefore it
is important to study quasilinear spaces with group structure. Quasilinear spaces
with group structure have remarkable algebraic properties (such as cancellation
law, existence of center, quasidistributive law, etc.) and can be effectively used for
algebraic calculations. Some early results related to quasilinear spaces can be found
in –, .
In this work we study the algebraic properties of intervals in the lines of ,
taking into account that intervals are quasilinear systems. We show that in a quasi-
linear space with group structure one can introduce analogues of the theory of
linear spaces, like linear combinations, linear dependence, basis, etc. The formu-
lation and solution of certain algebraic problems in quasilinear spaces is based on
an appropriate terminology and notation. We brieﬂy outline a theory of quasilinear
spaces with group structure, which is an extension of the theory of linear spaces.
Here we develop the theory using an alternative and more simple approach than
the one in , considering ﬁrst in some detail the symmetric case. Only a few