$\alpha$-filters and $\alpha$-order-ideals in distributive  quasicomplemented semilattices

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

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Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets

The Electronic Journal of Combinatorics ◽

10.37236/4334 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 11

Author(s):

Darij Grinberg ◽

Tom Roby

Keyword(s):

Finite Order ◽

Disjoint Union ◽

Parallel Analysis ◽

Birational Map ◽

Finite Poset ◽

Order Ideals ◽

Set Up ◽

Grafting Onto

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.

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Order ideals in a $C^\ast$ -algebra and its dual

Duke Mathematical Journal ◽

10.1215/s0012-7094-63-03042-4 ◽

1963 ◽

Vol 30 (3) ◽

pp. 391-411 ◽

Cited By ~ 56

Author(s):

Edward G. Effros

Keyword(s):

Order Ideals

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On estimating the number of order ideals in partial orders, with some applications

Journal of Statistical Planning and Inference ◽

10.1016/0378-3758(93)90012-u ◽

1993 ◽

Vol 34 (2) ◽

pp. 281-290 ◽

Cited By ~ 4

Author(s):

George Steiner

Keyword(s):

Partial Orders ◽

Order Ideals

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ALMOST FACTORIZABLE LOCALLY INVERSE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006832 ◽

2011 ◽

Vol 21 (07) ◽

pp. 1037-1052 ◽

Cited By ~ 2

Author(s):

MÁRIA B. SZENDREI

Keyword(s):

Inverse Semigroups ◽

Completely Simple Semigroups ◽

Order Ideals ◽

Locally Inverse Semigroups ◽

Completely Simple

We introduce a notion of almost factorizability within the class of all locally inverse semigroups by requiring a property of order ideals, and we prove that the almost factorizable locally inverse semigroups are just the homomorphic images of Pastijn products of normal bands by completely simple semigroups.

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The monomial conjecture and order ideals

Journal of Algebra ◽

10.1016/j.jalgebra.2013.03.006 ◽

2013 ◽

Vol 383 ◽

pp. 232-241 ◽

Cited By ~ 1

Author(s):

S.P. Dutta

Keyword(s):

Order Ideals

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ON AN ORDER-BASED CONSTRUCTION OF A TOPOLOGICAL GROUPOID FROM AN INVERSE SEMIGROUP

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091506000083 ◽

2008 ◽

Vol 51 (2) ◽

pp. 387-406 ◽

Cited By ~ 21

Author(s):

Daniel H. Lenz

Keyword(s):

Inverse Semigroup ◽

Inverse Semigroups ◽

Invariant Sets ◽

Ideal Structure ◽

Topological Groupoid ◽

Unit Space ◽

Order Ideals ◽

The Ideal

AbstractWe show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

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