Reliable Computing 7: 129–140, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Multiplication Distributivity of Proper and
EVGENIJA D. POPOVA
Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,
block 8, BG-1113 Soﬁa, Bulgaria, email: email@example.com
(Received: 30 April 1999; accepted: 19 October 1999)
Abstract. The arithmetic on an extended set of proper and improper intervals presents algebraic
completion of the conventional interval arithmetic allowing thus efﬁcient solution of some interval
algebraic problems. In this paper we summarize and present all distributive relations, known by now,
on multiplication and addition of generalized (proper and improper) intervals.
Among several extensions of the classical interval arithmetic that have been pro-
posed, we consider that one aiming at an algebraic completion of interval arithmetic.
The algebraic extension is developed by H.-J. Ortolf  and E. Kaucher , ,
further investigated by E. Garde
nes et al. , , S. Markov , , and others. The
set of normal (proper) intervals is extended by improper intervals and the interval
arithmetic operations and functions are extended correspondingly. The generalized
interval arithmetic structure, thus obtained, possesses group properties with respect
to addition and multiplication operations. Lattice operations are closed with respect
to the inclusion order relation. Handling of norm and metric are very similar to
norm and metric in linear spaces . In order to emphasize that a generalized
interval can be considered as a pair of a proper interval (in set-theoretical sense)
and a “direction”, sometimes the algebraic extension of the conventional interval
arithmetic is called directed interval arithmetic , . The term “modal interval
analysis”  reﬂects an interpretation of generalized intervals in terms of modal
The algebraic properties of the generalized interval arithmetic make it a suitable
environment for solving interval algebraic problems, e. g. some interval algebraic
equations, which are not linear in general, can be solved explicitly just by applying
elementary algebraic transformations due to the existence of inverse elements with
respect to addition and multiplication operations . However, the efﬁcient solu-
tion of some interval algebraic problems is hampered by the lack of well studied
distributive relations between generalized (proper and improper) intervals.
This work was supported by the Bulgarian National Science Fund under grant No. I-903/99.