354 research outputs found
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders
We investigate the ordinal invariants height, length, and width of well quasi
orders (WQO), with particular emphasis on width, an invariant of interest for
the larger class of orders with finite antichain condition (FAC). We show that
the width in the class of FAC orders is completely determined by the width in
the class of WQOs, in the sense that if we know how to calculate the width of
any WQO then we have a procedure to calculate the width of any given FAC order.
We show how the width of WQO orders obtained via some classical constructions
can sometimes be computed in a compositional way. In particular, this allows
proving that every ordinal can be obtained as the width of some WQO poset. One
of the difficult questions is to give a complete formula for the width of
Cartesian products of WQOs. Even the width of the product of two ordinals is
only known through a complex recursive formula. Although we have not given a
complete answer to this question we have advanced the state of knowledge by
considering some more complex special cases and in particular by calculating
the width of certain products containing three factors. In the course of
writing the paper we have discovered that some of the relevant literature was
written on cross-purposes and some of the notions re-discovered several times.
Therefore we also use the occasion to give a unified presentation of the known
results
On the Duality of Semiantichains and Unichain Coverings
We study a min-max relation conjectured by Saks and West: For any two posets
and the size of a maximum semiantichain and the size of a minimum
unichain covering in the product are equal. For positive we state
conditions on and that imply the min-max relation. Based on these
conditions we identify some new families of posets where the conjecture holds
and get easy proofs for several instances where the conjecture had been
verified before. However, we also have examples showing that in general the
min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure
Combinatorial symbolic powers
Symbolic powers are studied in the combinatorial context of monomial ideals.
When the ideals are generated by quadratic squarefree monomials, the generators
of the symbolic powers are obstructions to vertex covering in the associated
graph and its blowups. As a result, perfect graphs play an important role in
the theory, dual to the role played by perfect graphs in the theory of secants
of monomial ideals. We use Gr\"obner degenerations as a tool to reduce
questions about symbolic powers of arbitrary ideals to the monomial case. Among
the applications are a new, unified approach to the Gr\"obner bases of symbolic
powers of determinantal and Pfaffian ideals.Comment: 29 pages, 3 figures, Positive characteristic results incorporated
into main body of pape
Transversal Lattices
A flat of a matroid is cyclic if it is a union of circuits; such flats form a
lattice under inclusion and, up to isomorphism, all lattices can be obtained
this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic
flats are isomorphic to it are transversal. We investigate some sufficient
conditions for a lattice to be a Tr-lattice; a corollary is that distributive
lattices of dimension at most two are Tr-lattices. We give a necessary
condition: each element in a Tr-lattice has at most two covers. We also give
constructions that produce new Tr-lattices from known Tr-lattices.Comment: 12 pages; 5 figure
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
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