Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.
A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). Geometric homomorphisms can be used to define a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. In this paper, the homomorphism poset of K_{3,3} is determined.
Let ${\cal H}(P,C_3)$ be the set of order homomorphisms from a poset $P$ to the chain $C_3 = 1 < 2 < 3$. We develop a recursive approach for the calculation of the cardinality of ${\cal H}(P,C_3)$, and we apply it on several types of posets, including $P = C_3 \times C_3 \times C_k$ and $P = {\cal H}(C_k, C_3)$; for the latter poset $P$, we derive a direct formula for $\# {\cal H} ( P, C_3 )$.
Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P \in \mathfrak{P}_r$, a one-to-one mapping $ρ_P : {\cal H}(P,R) \rightarrow {\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R \sqsubseteq_G S$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite posets $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. In consequence…
We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable and we compute their homotopy type. Our method rests on Bibby's description of such posets by means of "labeled partitions": after giving an EL-labeling and counting homology chains for general posets of labeled partitions, we obtain the stated results by considering the appropriate subposets.
We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of…
The $P$-partition generating function of a (naturally labeled) poset $P$ is a quasisymmetric function enumerating order-preserving maps from $P$ to $\mathbb{Z}^+$. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the $P$-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating fu…
In this paper, we first give formulas for the order polynomial $Ω(\Pw; t)$ and the Eulerian polynomial $e(\Pw; λ)$ of a finite labeled poset $(P, ω)$ using the adjacency matrix of what we call the $ω$-graph of $(P, ω)$. We then derive various recursion formulas for $Ω(\Pw; t)$ and $e(\Pw; λ)$ and discuss some applications of these formulas to Bernoulli numbers and Bernoulli polynomials. Finally, we give a recursive algorithm using a single linear operator on a vector space. This algorithm provides a uniform method to construct a family of new invariants for labeled posets $(\Pw)$, which includ…
Let $\mathrm{XY_{L,T}}$ denote the class of countably infinite $L$-structures that satisfy the axioms $T$ and in which all homomorphisms of type X (these could be homomorphisms, monomorphisms, or isomorphisms) between finite substructures of $M$ are restrictions of an endomorphism of $M$ of type Y (for example, an automorphism or a surjective endomorphism). Lockett and Truss introduced 18 such morphism-extension classes for relational structures. For a given pair $L,T$, however, two or more morphism-extension properties may define the same class of structures. In this paper, we establish all e…
Given finite posets $P$ and $Q$, we consider a specific ideal $L(P,Q)$, whose minimal monomial generators correspond to order-preserving maps $ϕ:P\rightarrow Q$. We study algebraic invariants of those ideals. In particular, sharp lower and upper bounds for the Castelnuovo-Mumford regularity and the projective dimension are provided. Precise formulas are obtained for a large subclass of these ideals. Moreover, we provide complete characterizations for several algebraic properties of $L(P,Q)$, including being Buchsbaum, Cohen-Macaulay, Gorenstein and having a linear resolution. We also give a pa…
A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order.
To every labeled poset (P,ω), one can associate a quasisymmetric generating function for its (P,ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of line…
We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.
Given a finite graded poset with labeled Hasse diagram, we construct a quasi- symmetric generating function for (saturated) chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives a Hopf morphism from an incidence algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions.
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.
The $(P, ω)$-partition generating function of a labeled poset $(P, ω)$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to $\mathbb{Z}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{ψ_α\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients…
In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this…
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…
Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to sphe…
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.
We describe a Markov-Chain-Monte-Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one's choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n=40 or so do…
Given a finite poset $P$, we associate a simple graph denoted by $G_P$ with all connected order ideals of $P$ as vertices, and two vertices are adjacent if and only if they have nonempty intersection and are incomparable with respect to set inclusion. We establish a bijection between the set of maximum independent sets of $G_P$ and the set of $P$-forests, introduced by Féray and Reiner in their study of the fundamental generating function $F_P(\textbf{x})$ associated with $P$-partitions. Based on this bijection, in the cases when $P$ is naturally labeled we show that $F_P(\textbf{x})$ can fact…
