Related papers: Lattices from Pointed Building Sets: Generalized O…
A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…
The Sasaki projection and its dual were introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. In a previous paper the authors showed that the Sasaki operations induced by the Sasaki…
In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…
In this paper we consider a general way of constructing profinite struc- tures based on a given framework - a countable family of objects and a countable family of recognisers (e.g. formulas). The main theorem states: A subset of a family…
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of…
The Cambrian lattices, introduced in (Reading, 2006), generalize the Tamari lattice to any choice of Coxeter element in any finite Coxeter group. They are further generalized to the m-Cambrian lattices (Stump, Thomas, Williams, 2015).…
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
For a tensor triangulated category which is well generated in the sense of Neeman, it is shown that the collection of Bousfield classes forms a set. This set has a natural structure of a complete lattice which is then studied, using the…
Let X be a right-angled building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group…
We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.
We explore lattice structures on integer binary relations (i.e. binary relations on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$) and on integer posets (i.e. partial orders on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$).…
We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice…
In 1986, Oliver Pretzel studied the set of orientations of a connected finite graph $G$ and showed that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of…
A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between…
We explore some of the properties of a subposet of the Tamari lattice introduced by Pallo, which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within…
We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated…
Laminations are a combinatorial and topological way to study Julia sets. Laminations give information about the structure of parameter space of degree $d$ polynomials with connected Julia sets. We first study fixed point portraits in…
For positive integers $m$ and $k$, we introduce a family of lattices $\mathcal{C}_{k}^{(m)}$ associated to the Cambrian lattice $\mathcal{C}_{k}$ of the dihedral group $I_{2}(k)$. We show that $\mathcal{C}_{k}^{(m)}$ satisfies some basic…
We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not…
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and…