57 research outputs found
(m,n)-Semirings and a Generalized Fault Tolerance Algebra of Systems
We propose a new class of mathematical structures called (m,n)-semirings}
(which generalize the usual semirings), and describe their basic properties. We
also define partial ordering, and generalize the concepts of congruence,
homomorphism, ideals, etc., for (m,n)-semirings. Following earlier work by Rao,
we consider a system as made up of several components whose failures may cause
it to fail, and represent the set of systems algebraically as an
(m,n)-semiring. Based on the characteristics of these components we present a
formalism to compare the fault tolerance behaviour of two systems using our
framework of a partially ordered (m,n)-semiring.Comment: 26 pages; extension of arXiv:0907.3194v1 [math.GM
Congruence-simple semirings without nilpotent elements
We provide a classification of congruence-simple semirings with a
multiplicatively absorbing element and without non-trivial nilpotent elements
On generators of commutative semifields
We study ideal-simple commutative semirings and summarize the results giving
their classification, in particular when they are finitely generated. In the
principal case of (para)semifields, we then consider their minimal number of
generators and show that it grows linearly with the depth of an associated
rooted forest.Comment: 12 page
Semifields and a theorem of Abhyankar
summary:Abhyankar proved that every field of finite transcendence degree over or over a finite field is a homomorphic image of a subring of the ring of polynomials (for some depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings
The semiring of 1-preserving endomorphisms of a semilattice
summary:We prove that the semirings of 1-preserving and of 0,1-preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way
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