9,599 research outputs found
An Algorithmic Metatheorem for Directed Treewidth
The notion of directed treewidth was introduced by Johnson, Robertson,
Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as
a first step towards an algorithmic metatheory for digraphs. They showed that
some NP-complete properties such as Hamiltonicity can be decided in polynomial
time on digraphs of constant directed treewidth. Nevertheless, despite more
than one decade of intensive research, the list of hard combinatorial problems
that are known to be solvable in polynomial time when restricted to digraphs of
constant directed treewidth has remained scarce. In this work we enrich this
list by providing for the first time an algorithmic metatheorem connecting the
monadic second order logic of graphs to directed treewidth. We show that most
of the known positive algorithmic results for digraphs of constant directed
treewidth can be reformulated in terms of our metatheorem. Additionally, we
show how to use our metatheorem to provide polynomial time algorithms for two
classes of combinatorial problems that have not yet been studied in the context
of directed width measures. More precisely, for each fixed , we show how to count in polynomial time on digraphs of directed
treewidth , the number of minimum spanning strong subgraphs that are the
union of directed paths, and the number of maximal subgraphs that are the
union of directed paths and satisfy a given minor closed property. To prove
our metatheorem we devise two technical tools which we believe to be of
independent interest. First, we introduce the notion of tree-zig-zag number of
a digraph, a new directed width measure that is at most a constant times
directed treewidth. Second, we introduce the notion of -saturated tree slice
language, a new formalism for the specification and manipulation of infinite
sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic
Right Coideal Subalgebras of the Quantum Borel Algebra of type G2
In this paper we describe the right coideal subalgebras containing all
group-like elements of the multiparameter quantum group Uq+(g), where g is a
simple Lie algebra of type G2, while the main parameter of quantization q is
not a root of 1. If the multiplicative order t of q is finite, t>4, t different
from 6, then the same classification remains valid for homogeneous right
coideal subalgebras of the positive part uq+(g) of the multiparameter version
of the small Lusztig quantum group
On monoids, 2-firs, and semifirs
Several authors have studied the question of when the monoid ring DM of a
monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir,
or a 2-fir (definitions recalled in section 1). It is known that for M
nontrivial, a necessary condition for any of these properties to hold is that D
be a division ring. Under that assumption, necessary and sufficient conditions
on M are known for DM to be a right or left fir, and various conditions on M
have been proved necessary or sufficient for DM to be a 2-fir or semifir.
A sufficient condition for DM to be a semifir is that M be a direct limit of
monoids which are free products of free monoids and free groups. W.Dicks has
conjectured that this is also necessary. However F.Ced\'o has given an example
of a monoid M which is not such a direct limit, but satisfies the known
necessary conditions for DM to be a semifir. It is an open question whether for
this M, the rings DM are semifirs.
We note some reformulations of the known necessary conditions for DM to be a
2-fir or a semifir, motivate Ced\'o's construction and a variant, and recover
Ced\'o's results for both constructions.
Any homomorphism from a monoid M into \Z induces a \Z-grading on DM, and we
show that for the two monoids in question, the rings DM are "homogeneous
semifirs" with respect to all such nontrivial \Z-gradings; i.e., have (roughly)
the property that every finitely generated homogeneous one-sided ideal is free.
If M is a monoid such that DM is an n-fir, and N a "well-behaved" submonoid
of M, we obtain results on DN. Using these, we show that for M a monoid such
that DM is a 2-fir, mutual commutativity is an equivalence relation on
nonidentity elements of M, and each equivalence class, together with the
identity element, is a directed union of infinite cyclic groups or infinite
cyclic monoids.
Several open questions are noted.Comment: 28 pages. To appear, Semigroup Forum. Some clarifications and
corrections from previous versio
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