876 research outputs found
Exact rings and semirings
We introduce and study an abstract class of semirings, which we call exact
semirings, defined by a Hahn-Banach-type separation property on modules. Our
motivation comes from the tropical semiring, and in particular a desire to
understand the often surprising extent to which it behaves like a field. The
definition of exactness abstracts an elementary property of fields and the
tropical semiring, which we believe is fundamental to explaining this
similarity. The class of exact semirings turns out to include many other
important examples of both rings (proper quotients of principal ideal domains,
matrix rings and finite group rings over these and over fields), and semirings
(the Boolean semiring, generalisations of the tropical semiring, matrix
semirings and group semirings over these).Comment: 17 pages; fixed typos, clarified a few points, changed notation in
Example 6.
Green's J-order and the rank of tropical matrices
We study Green's J-order and J-equivalence for the semigroup of all n-by-n
matrices over the tropical semiring. We give an exact characterisation of the
J-order, in terms of morphisms between tropical convex sets. We establish
connections between the J-order, isometries of tropical convex sets, and
various notions of rank for tropical matrices. We also study the relationship
between the relations J and D; Izhakian and Margolis have observed that for the semigroup of all 3-by-3 matrices over the tropical semiring with
, but in contrast, we show that for all full matrix semigroups
over the finitary tropical semiring.Comment: 21 pages, exposition improve
Tropical Cramer Determinants Revisited
We prove general Cramer type theorems for linear systems over various
extensions of the tropical semiring, in which tropical numbers are enriched
with an information of multiplicity, sign, or argument. We obtain existence or
uniqueness results, which extend or refine earlier results of Gondran and
Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and
Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also
discussed; in particular, some of our proofs lead to Jacobi and Gauss-Seidel
type algorithms to solve linear systems in suitably extended tropical
semirings.Comment: 41 pages, 5 Figure
Recommended from our members
Approximate comparison of distance automata
Distance automata are automata weighted over the semiring (N∪ {∞}, min,+) (the tropical semiring). Such automata compute functions from words to N
∪{∞} such as the number of occurrences of a given letter. It is known that testing f 0 and two functions f,g computed by distance automata, answers "yes" if f <= (1-ε ) g, "no" if f \not\leq g, and may answer "yes" or "no" in all other cases. This result highly refines previously known decidability results of the same type. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring. We also provide another theorem, of affine domination, which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata
Diameters of commuting graphs of matrices over semirings
We calculate the diameters of commuting graphs of matrices over the binary
Boolean semiring, the tropical semiring and an arbitrary nonentire commutative
semiring. We also find the lower bound for the diameter of the commuting graph
of the semiring of matrices over an arbitrary commutative entire antinegative
semiring.Comment: 8 page
- …