6,403 research outputs found
On the homomorphism order of labeled posets
Partially ordered sets labeled with k labels (k-posets) and their
homomorphisms are examined. We give a representation of directed graphs by
k-posets; this provides a new proof of the universality of the homomorphism
order of k-posets. This universal order is a distributive lattice. We
investigate some other properties, namely the infinite distributivity, the
computation of infinite suprema and infima, and the complexity of certain
decision problems involving the homomorphism order of k-posets. Sublattices are
also examined.Comment: 14 page
Complexity of matrix problems
In representation theory, the problem of classifying pairs of matrices up to
simultaneous similarity is used as a measure of complexity; classification
problems containing it are called wild problems. We show in an explicit form
that this problem contains all classification matrix problems given by quivers
or posets. Then we prove that it does not contain (but is contained in) the
problem of classifying three-valent tensors. Hence, all wild classification
problems given by quivers or posets have the same complexity; moreover, a
solution of any one of these problems implies a solution of each of the others.
The problem of classifying three-valent tensors is more complicated.Comment: 24 page
EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets
We show that a finite graded lattice of rank n is supersolvable if and only
if it has an EL-labeling where the labels along any maximal chain form a
permutation. We call such a labeling an S_n EL-labeling and we consider finite
graded posets of rank n with unique top and bottom elements that have an S_n
EL-labeling. We describe a type A 0-Hecke algebra action on the maximal chains
of such posets. This action is local and gives a representation of these Hecke
algebras whose character has characteristic that is closely related to
Ehrenborg's flag quasi-symmetric function. We ask what other classes of posets
have such an action and in particular we show that finite graded lattices of
rank n have such an action if and only if they have an S_n EL-labeling.Comment: 18 pages, 8 figures. Added JCTA reference and included some minor
corrections suggested by refere
A representation for the modules of a graph and applications
We describe a simple representation for the modules of a graph C. We show that the modules of C are in one-to-one correspondence with the ideaIs of certain posets. These posets are characterizaded and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate alI modules of C, (ii) count the number of modules of C, (iii) find a maximal module satisfying some hereditary property of C and (iv) find a connected non-trivial module of C
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