Reliable Computing 3: 199–207, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Isomorphic Embeddings of Abstract Interval
SVETOSLAV M. MARKOV
Institute of Mathematics and Computer Sciences, Bulgarian Academy of Sciences,
“Acad. G. Bonchev” st., block 8, 1113 Soﬁa, Bulgaria, e-mail: firstname.lastname@example.org
(Received: 27 November 1996; accepted: 2 March 1997)
Abstract. We study new abstract algebraic systems generalizing the system of real compact intervals
with addition and multiplication by scalar and the isomorphic embedding of these systems into
systems having group properties with respect to addition.
We study the algebraic relations between: i) extended interval arithmetic over
normal (proper) intervals using inner (nonstandard) arithmetic operations , ,
and, ii) extended interval arithmetic using improper intervals , , . We
have shown in  that the ﬁrst system is a “projection” of the second one on
the set of proper intervals. Here we continue our work from  aiming to show
that the second system is an isomorphic algebraic extension of the ﬁrst one. We
study the essential algebraic properties of the system of intervals necessary for such
isomorphic embedding. To this end we deliberately exclude from consideration
the inclusion relation between intervals with the corresponding lattice operations
involved, concentrating on the properties of the operations addition, subtraction
and multiplication by scalar. A similar approach has been used in , where more
general systems (e.g. convex compact sets in IR
) are studied. An essential difference
between our work and  is that we are able to construct isomorphic embeddings,
whereas the embedding in  is not isomorphic. To achieve an isomorphism we ﬁrst
extend the quasilinear space  by means of a complete second distributivity law,
which involves negation and inner addition , . The modiﬁed structure thus
obtained, called extended quasilinear system, is isomorphically embedded in an
analogue of a linear system, called extended linear system, having group properties
with respect to addition. Particular systems of intervals which are extended groups
and extended linear systems have been considered in ,  and other sources;
here we give an abstract algebraic theory of these systems.
In Section 2 we shortly repeat some of the basic concepts from , simplifying
the deﬁnition of extended semigroup (e.g. we do not require here uniqueness
Partially supported by the National Science Fund under contract No. I 507/95. The author is
indebted to the referee for his numerous suggestions.