686 research outputs found
Semigroups of tolerance relations
AbstractWe give an abstract algebraic characterization of semigroups of tolerance relations and semigroups of symmetric binary relations
Varieties whose tolerances are homomorphic images of their congruences
The homomorphic image of a congruence is always a tolerance (relation) but,
within a given variety, a tolerance is not necessarily obtained this way. By a
Maltsev-like condition, we characterize varieties whose tolerances are
homomorphic images of their congruences (TImC). As corollaries, we prove that
the variety of semilattices, all varieties of lattices, and all varieties of
unary algebras have TImC. We show that a congruence n-permutable variety has
TImC if and only if it is congruence permutable, and construct an idempotent
variety with a majority term that fails TImC
Tolerances as images of congruences in varieties defined by linear identities
An identity s=t is linear if each variable occurs at most once in each of the
terms s and t. Let T be a tolerance relation of an algebra A in a variety
defined by a set of linear identities. We prove that there exist an algebra B
in the same variety and a congruence Theta of B such that a homomorphism from B
onto A maps Theta onto T.Comment: 3 page
(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
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