Lattice Of Order Ideals Research Articles

Graded Steinberg algebras and their representations

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite $k$-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.

A Refinement of Weak Order Intervals into Distributive Lattices

In this paper we consider arbitrary intervals in the left weak order on the symmetric group Sn. We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least \({\left(\lfloor {\frac{n}{2}} \rfloor \right)!}\) permutations in Sn that form a rank-symmetric interval in the weak order.

Abstract root systems and intervals in weak Bruhat order

Abstract We present a new axiomatization of what is known as the “root system” of a Coxeter group that does not involve vector spaces or Coxeter groups. We use this to combinatorially characterize the structure of intervals in weak Bruhat order in much the same way that finite distributive lattices are characterized as the lattices of order ideals of partially ordered sets. In fact, the result for distributive lattices follows as a special case.

A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or −1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. We prove product formulas counting these order ideals and give the rank generating function of some of the corresponding lattices of order ideals. We also prove an expansion of the tournament generating function as a sum over TSSCPPs. This result is analogous to a result of Robbins and Rumsey expanding the tournament generating function as a sum over alternating sign matrices.

The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind–MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. In particular, if K(P) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.

Lattices of order ideals on ordered semigroups

Lattices of order ideals on ordered semigroups