We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fraïssé sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of lattices associated with it. Each of these two sequences has a limit object -- an inverse limit, which is an object of our interest as well.
In the first chapter there are some preliminaries considering partial orders, lattices, topology, inverse limits, category theory and Fraïssé theory, which are used later. In the second chapter there are our results considering a Fraïssé sequence in category of finite posets with quotient maps and properties of inverse limit of this sequence. In the third chapter we investigate connections between posets and order ideals corresponding to them, getting an inductive sequence made of these ideals; then we study properties of the inverse limit of this sequence.
We introduce a functor ${\sf As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction ${\sf As}$ provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from ${\sf As}$ are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combin…
We consider profunctors $f : P \promap Q$ between posets and introduce their {\em graph} and {\em ascent}. The profunctors $\Pro(P,Q)$ form themselves a poset, and we consider a partition $\cI \sqcup \cF$ of this into a down-set $\cI$ and up-set $\cF$, called a {\it cut}. To elements of $\cF$ we associate their graphs, and to elements of $\cI$ we associate their ascents. Our basic result is that this, suitable refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of $Q \times P$. Cuts in finite Booleans lattices correspond precisely to finite simp…
In this paper, we first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then we construct the corresponding nonassociative permutative coproduct, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and is a co-free non-associative permutative coalgebra. In the end, we give an explicit duality between the non-associative permutative product and the proposed non-associative permutative coproduct. Finally, we prove that the results in this paper remain true for the fini…
For any finite poset $P$ we have the poset of isotone maps $\text{Hom}(P,\mathbb{N})$, also called $P^{op}$-partitions. To any poset ideal ${\mathcal J}$ in $\text{Hom}(P,\mathbb{N})$, finite or infinite, we associate monomial ideals: the letterplace ideal $L({\mathcal J},P)$ and the Alexander dual co-letterplace ideal $L(P,{\mathcal J})$, and study them. We derive a class of monomial ideals in $k[x_p, p \in P]$ called $P$-stable. When $P$ is a chain we establish a duality on strongly stable ideals. We study the case when ${\mathcal J}$ is a principal poset ideal. When $P$ is a chain we constr…
The combined work of Bousquet-Mélou, Claesson, Dukes, Jelínek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent s…
We introduce the notion of a \emph{Whitney dual} of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge and chain-edge labelings which we call \emph{Whitney labelings}. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets. As applications of our main theorem, we show that geometric lattices, the lattice of noncrossi…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the botto…
We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey's rule for quantum affine algebras $U_q(E_{6,7,8}^{(1)})$ and partial information of denominator formulas for $U_q(E_{6,7,8}^{(1)})$…
There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, an…
Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P…
A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that $\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$…
A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P, P'$, let the \textit{poset Ramsey number} $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. Axenovich and Walzer introduced this concept in \textit{Order} (2017), where they proved $R(Q_2, Q_n) \le 2n + 2$ and $R(Q_n, Q_m) \le mn + n + m$, wher…
We introduce a new combinatorial condition on a subinterval of a poset P (a clamped subinterval) that allows us to relate the Auslander-Reiten quiver of the bounded derived category of P to that of the subinterval. Applications include the determination of when a poset is fractionally Calabi-Yau and the computation of the Auslander-Reiten quivers of both the bounded derived category and of the module category.
We continue the Coxeter spectral analysis of finite connected posets $I$ that are non-negative in the sense that their symmetric Gram matrix $G_I:=\frac{1}{2}(C_I + C_I^{tr})\in\mathbb{M}_{m}(\mathbb{Q})$ is positive semi-definite of rank $n\geq 0$, where $C_I\in\mathbb{M}_m(\mathbb{Z})$ is the incidence matrix of $I$ encoding the relation $\preceq_I$. We extend the results of [Fundam. Inform., 139.4(2015), 347--367] and give a complete Coxeter spectral classification of finite connected posets $I$ of Dynkin type $\mathbb{A}_n$. We show that such posets $I$, with $|I|>1$, yield exactly $\lfloo…
We analyze the poset of moves in chip-firing, as defined by Klivans and Liscio. Answering a question of Propp, we show that the move poset forms the join-irreducibles of the poset of configurations. The proof involves a graph augmentation and an analysis of configurations in which only one firing move is available. We then use this framework to analyze the problem of chip-firing on a line, where the move poset is relevant to the problem of labeled chip-firing.
We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.
The gold partition conjecture and the Lexicographic sum of posets
Yonsei Univeristy（延世大学）；Kyungpook National University（庆北国立大学）；Thomas J. Watson Research Center, IBM（IBM 托马斯·J·沃森研究中心）；Pusan National University（釜山国立大学）
In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal ``ideal solutions'' are to be generated. This in particular allows us to study the complexity of the problem un…
We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of semi-simple Lie algebras. The faces of these polyhedra correspond to certain partitions of the underlying poset and we give a combinatorial characterization of these partitions. We specify a class of marked posets that give rise to polyhedra with facets in correspondence to the covering relations of the poset. On the convex geometrical side, we describe the recession…
