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Abstract:
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Semiring with zero and identity is an algebraic structure which generalizes a ring. Namely,
while a ring under addition is a group, a semiring is only a monoid. The lack of substraction
makes this structure far more difficult for investigation than a ring.
The subject of investigation in this thesis are matrices over commutative semirings
(wiht zero and identity). Motivation for this study is contained in an attempt to determine
which properties for matrices over commutative rings may be extended to matrices over
commutative semirings, and, also, which is closely connected to this question, how can
the properties of modules over rings be extended to semimodules over semirings.
One may distinguish three types of the obtained results.
First, the known results concerning dimension of spaces of n-tuples of elements from
a semiring are extended to a new class of semirings from the known ones until now, and
some errors from a paper by other authors are corrected. This question is closely related
to the question of invertibility of matrices over semirings.
Second type of results concerns investigation of zero divisors in a semiring of all
matrices over commutative semirings, in particular for a class of inverse semirings (which
are those semirings for which there exists some sort of a generalized inverse with respect
to addition). Because of the lack of substraction, one cannot use the determinant, as in the
case of matrices over commutative semirings, but, because of the fact that the semirings
in question are inverse semirings, it is possible to define some sort of determinant in this
case, which allows the formulation of corresponding results in this case. It is interesting
that for a class of matrices for which the results are obtained, left and right zero divisors
may differ, which is not the case for commutative rings.
The third type of results is about the question of introducing a new rank for matrices
over commutative semirings. For such matrices, there already exists a number of rank
functions, generalizing the rank function for matrices over fields. In this thesis, a new
rank function is proposed, which is based on the permanent, which is possible to define
for semirings, unlike the determinant, and which has good enough properties to allow a
definition of rank in such a way. |