1,210 research outputs found
On lattices and their ideal lattices, and posets and their ideal posets
For P a poset or lattice, let Id(P) denote the poset, respectively, lattice,
of upward directed downsets in P, including the empty set, and let
id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that
Id(P) is always, and id(P) often, "essentially larger" than P. In the first
vein, we find that a poset P admits no "<"-respecting map (and so in
particular, no one-to-one isotone map) from Id(P) into P, and, going the other
way, that an upper semilattice S admits no semilattice homomorphism from any
subsemilattice of itself onto Id(S).
The slightly smaller object id(P) is known to be isomorphic to P if and only
if P has ascending chain condition. This result is strengthened to say that the
only posets P_0 such that for every natural number n there exists a poset P_n
with id^n(P_n)\cong P_0 are those having ascending chain condition. On the
other hand, a wide class of cases is noted here where id(P) is embeddable in P.
Counterexamples are given to many variants of the results proved.Comment: 8 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. After publication, updates, errata,
etc. may be noted at that pag
Morphisms and order ideals of toric posets
Toric posets are cyclic analogues of finite posets. They can be viewed
combinatorially as equivalence classes of acyclic orientations generated by
converting sources into sinks, or geometrically as chambers of toric graphic
hyperplane arrangements. In this paper we study toric intervals, morphisms, and
order ideals, and we provide a connection to cyclic reducibility and conjugacy
in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as
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