109,788 research outputs found
Contiguity orders
This paper is devoted to the study of contiguity orders i.e. orders having a linear extension extension L such that all upper (or lower) cover sets are intervals of L. This new class is a strict generalization of both interval orders and N-free orders and is linearly recognizable. It is proved that computing the number of contiguity extensions is #P-complete and that the dimension of height one contiguity orders is polynomially tractable. Moreover the membership is a comparability invariant on bi-contiguity orders. Finally for strong-contiguity orders the calculation of the dimension is NP-complete
Order Invariance on Decomposable Structures
Order-invariant formulas access an ordering on a structure's universe, but
the model relation is independent of the used ordering. Order invariance is
frequently used for logic-based approaches in computer science. Order-invariant
formulas capture unordered problems of complexity classes and they model the
independence of the answer to a database query from low-level aspects of
databases. We study the expressive power of order-invariant monadic
second-order (MSO) and first-order (FO) logic on restricted classes of
structures that admit certain forms of tree decompositions (not necessarily of
bounded width).
While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO
with modulo-counting predicates), we show that order-invariant MSO and CMSO are
equally expressive on graphs of bounded tree width and on planar graphs. This
extends an earlier result for trees due to Courcelle. Moreover, we show that
all properties definable in order-invariant FO are also definable in MSO on
these classes. These results are applications of a theorem that shows how to
lift up definability results for order-invariant logics from the bags of a
graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201
Computing generators of free modules over orders in group algebras
Let E be a number field and G be a finite group. Let A be any O_E-order of
full rank in the group algebra E[G] and X be a (left) A-lattice. We give a
necessary and sufficient condition for X to be free of given rank d over A. In
the case that the Wedderburn decomposition of E[G] is explicitly computable and
each component is in fact a matrix ring over a field, this leads to an
algorithm that either gives an A-basis for X or determines that no such basis
exists.
Let L/K be a finite Galois extension of number fields with Galois group G
such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to
certain Galois modules that arise naturally in this situation. For example, one
can take X to be O_L, the ring of algebraic integers of L, and A to be the
associated order A of O_L in E[G]. The application of the algorithm to this
special situation is implemented in Magma under certain extra hypotheses when
K=E=Q.Comment: 17 pages, latex, minor revision
On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results
Let L/K be a finite Galois extension of number fields with Galois group G.
Let p be a rational prime and let r be a non-positive integer. By examining the
structure of the p-adic group ring Z_p[G], we prove many new cases of the
p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair
(h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other
conjectures concerning the vanishing of certain elements in relative algebraic
K-groups. We then prove a conjecture of Burns concerning the annihilation of
class groups as Galois modules for a wide class of interesting extensions,
including cases in which the full ETNC in not known. Similarly, we construct
annihilators of higher dimensional algebraic K-groups of the ring of integers
in L.Comment: 33 pages, error in section 3.4 corrected. To appear in Transactions
of the AM
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