Manfred Knebusch
Monoid Valuations and Value Ordered Supervaluations.
Fri, 04/15/2011 - 15:01 — SomNambulAuthors: Zur Izhakian, Louis Rowen, Manfred Knebusch
Subjects: Commutative Algebra
AbstractWe complement two papers on supertropical valuation theory ([IKR1],[IKR2]) by
providing natural examples of m-valuations (= monoid valuations), after that of
supervaluations and transmissions between them. The supervaluations discussed
have values in totally ordered supertropical semirings, and the transmissions
discussed respect the orderings. Basics of a theory of such semirings and
transmissions are developed as far as needed.Supertropical linear algebra.
Tue, 08/03/2010 - 15:01 — SomNambulAuthors: Zur Izhakian, Louis Rowen, Manfred Knebusch
Subjects: Commutative Algebra
AbstractThe objective of this paper is to lay out the algebraic theory of
supertropical vector spaces and linear algebra, utilizing the key antisymmetric
relation of ``ghost surpasses.''Special attention is paid to the various
notions of ``base,'' which include d-base and s-base, and these are compared to
other treatments in the tropical theory. Whereas the number of elements in a
d-base may vary according to the d-base, it is shown that when an s-base
exists, it is unique up to permutation and multiplication by scalars, and can
be identified with a set of ``critical'' elements.Supertropical semirings and supervaluations.
Fri, 03/05/2010 - 16:01 — SomNambulAuthors: Zur Izhakian, Louis Rowen, Manfred Knebusch
Subjects: Commutative Algebra
AbstractWe interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so
called bipotent semiring $M$ (the usual max-plus setting), and then define a
\textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring
$U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U
\to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural
ordering which makes it a complete lattice. In the case that $R$ is a field,
hence for $v$ a Krull valuation, we give a complete explicit description of
$\Cov(v)$.Layered supertropical domains.
Wed, 12/09/2009 - 16:01 — SomNambulAuthors: Zur Izhakian, Louis Rowen, Manfred Knebusch
Subjects: Commutative Algebra
AbstractGeneralizing supertropical algebras, we consider a "layered" structure which
permits different ghost layers, and indicate how it is more amenable than the
unlayered construction to mathematical analysis, in particular with respect to
calculus. On the other hand, some of the matrix theory developed in [IR2] and
[IR4] is also developed in this more general setting.
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