980,266 research outputs found
Dagger closure in regular rings containing a field
We prove that dagger closure is trivial in regular domains containing a field
and that graded dagger closure is trivial in polynomial rings over a field. We
also prove that Heitmann's full rank one closure coincides with tight closure
in positive characteristic under some mild finiteness conditions. Furthermore,
we prove that dagger closure is always contained in solid closure and that the
forcing algebra for an element contained in dagger closure is parasolid.Comment: 12 pages, v2: added one corollary and two references, v3: Major
simplification in proof of main thm due to the suggestion of a referee, minor
changes in expositio
A reconstruction of the multipreference closure
The paper describes a preferential approach for dealing with exceptions in
KLM preferential logics, based on the rational closure. It is well known that
the rational closure does not allow an independent handling of the inheritance
of different defeasible properties of concepts. Several solutions have been
proposed to face this problem and the lexicographic closure is the most notable
one. In this work, we consider an alternative closure construction, called the
Multi Preference closure (MP-closure), that has been first considered for
reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure
in the propositional case and we show that it is a natural variant of Lehmann's
lexicographic closure. Abandoning Maximal Entropy (an alternative route already
considered but not explored by Lehmann) leads to a construction which exploits
a different lexicographic ordering w.r.t. the lexicographic closure, and
determines a preferential consequence relation rather than a rational
consequence relation. We show that, building on the MP-closure semantics,
rationality can be recovered, at least from the semantic point of view,
resulting in a rational consequence relation which is stronger than the
rational closure, but incomparable with the lexicographic closure. We also show
that the MP-closure is stronger than the Relevant Closure.Comment: 57 page
Entropy of Closure Operators
The entropy of a closure operator has been recently proposed for the study of
network coding and secret sharing. In this paper, we study closure operators in
relation to their entropy. We first introduce four different kinds of rank
functions for a given closure operator, which determine bounds on the entropy
of that operator. This yields new axioms for matroids based on their closure
operators. We also determine necessary conditions for a large class of closure
operators to be solvable. We then define the Shannon entropy of a closure
operator, and use it to prove that the set of closure entropies is dense.
Finally, we justify why we focus on the solvability of closure operators only.Comment: arXiv admin note: substantial text overlap with arXiv:1209.655
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