997 research outputs found
Variations on Algebra: monadicity and generalisations of equational theories
Dedicated to Rod Burstal
Bounds on the global dimension of certain piecewise hereditary categories
We give bounds on the global dimension of a finite length, piecewise
hereditary category in terms of quantitative connectivity properties of its
graph of indecomposables.
We use this to show that the global dimension of a finite dimensional,
piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of
a finite poset or more generally, a sincere algebra. This bound is tight.Comment: 7 pages; slightly revised; to appear in J. Pure Appl. Algebr
A Diffie-Hellman based key management scheme for hierarchical access control
All organizations share data in a carefully managed fashion\ud
by using access control mechanisms. We focus on enforcing access control by encrypting the data and managing the encryption keys. We make the realistic assumption that the structure of any organization is a hierarchy of security classes. Data from a certain security class can only be accessed by another security class, if it is higher or at the same level in the hierarchy. Otherwise access is denied. Our solution is based on the Die-Hellman key exchange protocol. We show, that the theoretical worst case performance of our solution is slightly better than that of all other existing solutions. We also show, that our performance in practical cases is linear in the size of the hierarchy, whereas the best results from the literature are quadratic
A coalgebraic semantics for causality in Petri nets
In this paper we revisit some pioneering efforts to equip Petri nets with
compact operational models for expressing causality. The models we propose have
a bisimilarity relation and a minimal representative for each equivalence
class, and they can be fully explained as coalgebras on a presheaf category on
an index category of partial orders. First, we provide a set-theoretic model in
the form of a a causal case graph, that is a labeled transition system where
states and transitions represent markings and firings of the net, respectively,
and are equipped with causal information. Most importantly, each state has a
poset representing causal dependencies among past events. Our first result
shows the correspondence with behavior structure semantics as proposed by
Trakhtenbrot and Rabinovich. Causal case graphs may be infinitely-branching and
have infinitely many states, but we show how they can be refined to get an
equivalent finitely-branching model. In it, states are equipped with
symmetries, which are essential for the existence of a minimal, often
finite-state, model. The next step is constructing a coalgebraic model. We
exploit the fact that events can be represented as names, and event generation
as name generation. Thus we can apply the Fiore-Turi framework: we model causal
relations as a suitable category of posets with action labels, and generation
of new events with causal dependencies as an endofunctor on this category. Then
we define a well-behaved category of coalgebras. Our coalgebraic model is still
infinite-state, but we exploit the equivalence between coalgebras over a class
of presheaves and History Dependent automata to derive a compact
representation, which is equivalent to our set-theoretical compact model.
Remarkably, state reduction is automatically performed along the equivalence.Comment: Accepted by Journal of Logical and Algebraic Methods in Programmin
Braids, posets and orthoschemes
In this article we study the curvature properties of the order complex of a
graded poset under a metric that we call the ``orthoscheme metric''. In
addition to other results, we characterize which rank 4 posets have CAT(0)
orthoscheme complexes and by applying this theorem to standard posets and
complexes associated with four-generator Artin groups, we are able to show that
the 5-string braid group is the fundamental group of a compact nonpositively
curved space.Comment: 33 pages, 16 figure
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