An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

A simple semidistributive lattice

Under broad finiteness conditions, such as the existence of a greatest or least element, a semidistributive lattice has the two element lattice as a homomorphic image, and so, if it has more than two elements, is not simple. However, the existence of a simple, semidistributive lattice with more than two elements has remained in question. This paper constructs such a lattice.

FROM JOIN-IRREDUCIBLES TO DIMENSION THEORY FOR LATTICES WITH CHAIN CONDITIONS

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J (L) of join-irreducible elements of L and the join-dependency relation DL on J (L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations [Formula: see text] As a consequence of this, we obtain the following results: Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom [Formula: see text] Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds: [Formula: see text] where ⊗ denotes the tensor product of commutative monoids.