# Morphisms and Order Ideals of Toric Posets

## Abstract

**:**

## 1. Introduction

- ■
- A set $C\subseteq V$ is a toric chain of $P(G,[\omega \left]\right)$ iff C is a chain of $P(G,{\omega}^{\prime})$ for all ${\omega}^{\prime}\in \left[\omega \right]$. (Proposition 5.3)
- ■
- The edge $\{i,j\}$ is in the toric transitive closure of $P(G,[\omega \left]\right)$ iff $\{i,j\}$ is in the transitive closure of $P(G,{\omega}^{\prime})$ for all ${\omega}^{\prime}\in \left[\omega \right]$. (Proposition 5.15)

- ■
- A partition $\pi \in {\mathsf{\Pi}}_{V}$ is a closed toric face partition of $P(G,[\omega \left]\right)$ iff π is a closed face partition of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$. (Theorem 4.7)
- ■
- A set $A\subseteq V$ is a (geometric) toric antichain of $P(G,[\omega \left]\right)$ iff A is an antichain of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$. (Proposition 5.17)
- ■
- If a set $I\subseteq V$ is a toric interval of $P(G,[\omega \left]\right)$, then I is an interval of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$. (Proposition 5.14)
- ■
- A set $J\subseteq V$ is a toric order ideal of $P(G,[\omega \left]\right)$ iff J is an order ideal of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$. (Proposition 7.3)
- ■
- Collapsing $P(G,[\omega \left]\right)$ by a partition $\pi \in {\mathsf{\Pi}}_{V}$ is a morphism of toric posets iff collapsing $P(G,{\omega}^{\prime})$ by π is a poset morphism for some ${\omega}^{\prime}\in \left[\omega \right]$. (Corollary 6.2)
- ■
- If an edge $\{i,j\}$ is in the Hasse diagram of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$, then it is in the toric Hasse diagram of $P(G,[\omega \left]\right)$. (Proposition 5.15)

## 2. Posets Geometrically

#### 2.1. Posets and Preposets

#### 2.2. Chambers of Hyperplane Arrangements

#### 2.3. Face Structure of Chambers

**Theorem 2.1.**

**Example 2.2.**

**Definition 2.3.**A set $F\subseteq {\mathbb{R}}^{V}$ is a closed face of the poset P if $F={\overline{F}}_{\pi}=\overline{c\left(P\right)}\cap {D}_{\pi}$ for some closed face partition $\pi ={cl}_{P}\left(\pi \right)$ of V. The interior of ${\overline{F}}_{\pi}$ with respect to the subspace topology of ${D}_{\pi}$ is called an open face of P, and denoted ${F}_{\pi}$. Let $Face\left(P\right)$ and $\overline{Face}\left(P\right)$ denote the set of open and closed faces of P, respectively. Finally, define the faces of the graphic arrangement $\mathcal{A}\left(G\right)$ to be the faces of the posets over G:

**Remark 2.4.**

**Example 2.5.**

- (i)
- as a unique orientation ${\omega}_{\pi}$ of G, where π is the partition into the strongly connected components;
- (ii)
- as a unique acyclic quotient $\omega /\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\sim}_{\pi}$ of an acyclic orientation $\omega \in Acyc\left(G\right)$.

## 3. Morphisms of Ordinary Posets

- combinatorially by the condition that $i{<}_{P}j$ is equivalent to $\varphi \left(i\right){<}_{{P}^{\prime}}\varphi \left(j\right)$ for all $i,j\in V$;
- geometrically by the equivalent condition that the induced isomorphism $\mathsf{\Phi}:{\mathbb{R}}^{V}\to {\mathbb{R}}^{{V}^{\prime}}$ maps $c\left(P\right)$ to $c\left({P}^{\prime}\right)$ bijectively.

#### 3.1. Quotient

#### 3.1.1. Contracting Partitions

**Remark 3.1.**

**Example 3.2.**

**Lemma 3.3.**

#### 3.1.2. Intervals and Antichains

**Definition 3.4.**

#### 3.2. Extension

#### 3.3. Inclusion

#### 3.4. Summary

- (i)
- quotient: Collapsing G by a partition π that preserves acyclicity of ω (projecting $c\left(P\right)$ to a flat ${D}_{\pi}$ of $\mathcal{A}\left({G}_{\pi}^{\prime}\right)$ for some closed partition $\pi ={cl}_{P}\left(\pi \right)$).
- (ii)
- inclusion: Adding vertices (adding dimensions).
- (iii)
- extension: Adding relations (intersecting with half-spaces).

## 4. Toric Posets and Preposets

#### 4.1. Toric Chambers and Posets

**Definition 4.1.**

- ■
- If ${x}_{i}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1\le {x}_{j}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1$, then include edge $i\to j$;
- ■
- If ${x}_{j}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1\le {x}_{i}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1$, then include edge $j\to i$.

**Definition 4.2.**

**Theorem 4.3.**

**Corollary 4.4.**

#### 4.2. Toric Faces and Preposets

**Definition 4.5.**

**Remark 4.6.**

**Theorem 4.7.**

**Lemma 4.8.**

- (a)
- If π is closed with respect to $P(G,\omega )$, then π is closed with respect to $P(G,[\omega \left]\right)$.
- (b)
- Closure is monotone: if $\pi {\le}_{V}{\pi}^{\prime}$, then ${cl}_{P}^{tor}\left(\pi \right){\le}_{V}{cl}_{P}^{tor}\left({\pi}^{\prime}\right)$.
- (c)
- If $\pi {\le}_{V}{\pi}^{\prime}{\le}_{V}{cl}_{P}^{tor}\left(\pi \right)$, then ${cl}_{P}^{tor}\left({\pi}^{\prime}\right)={cl}_{P}^{tor}\left(\pi \right)$.
- (d)
- ${cl}_{P(G,[\omega \left]\right)}^{tor}\left(\pi \right){\le}_{V}{cl}_{P(G,\omega )}\left(\pi \right)$.

**Proof.**

**Example 4.9.**

**Proof of Theorem 4.7.**

**Example 4.10.**

**Proposition 4.11.**

**Definition 4.12.**

**Example 4.13.**

## 5. Toric Intervals and Antichains

#### 5.1. Toric Total Orders

**Theorem 5.1.**

#### 5.2. Toric Directed Paths, Chains, and Transitivity

**Definition 5.2.**

**Proposition 5.3.**

- (a)
- C is a toric chain in P, with ${P|}_{C}=\left[({i}_{1},\dots ,{i}_{m})\right]$.
- (b)
- For every ${\omega}^{\prime}\in \left[\omega \right]$, the set C is a chain of $P(G,{\omega}^{\prime})$, ordered in some cyclic shift of $({i}_{1},\cdots ,{i}_{m})$.
- (c)
- For every ${\omega}^{\prime}\in \left[\omega \right]$, the set C occurs as a subsequence of a toric directed path in ${\omega}^{\prime}$, in some cyclic shift of the order $({i}_{1},\cdots ,{i}_{m})$.
- (d)
- Every total toric extension $\left[w\right]$ in ${\mathcal{L}}_{tor}\left(P\right)$ has the same restriction $\left[w{|}_{C}\right]=\left[({i}_{1},\dots ,{i}_{m})\right]$.

**Proposition 5.4.**

**Proposition 5.5.**

#### 5.3. Toric Hasse Diagrams

**Proposition 5.6.**

- (i)
- The edge $\{i,j\}$ is in the Hasse diagram, ${\widehat{G}}^{Hasse}\left(P\right)$.
- (ii)
- Removing ${H}_{i,j}$ enlarges the chamber $c\left(P\right)$.
- (iii)
- $\overline{c\left(P\right)}\cap {H}_{i,j}$ is a (closed) facet of P.
- (iv)
- The interval $[i,j]$ is precisely $\{i,j\}$.

**Remark 5.7.**

#### 5.4. Toric Intervals

**Definition 5.8.**

- (i)
- $I=\{k\in V:{x}_{i}\le {x}_{k}\le {x}_{j},\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in c\left(P\right)\}$;
- (ii)
- $I=\{k\in V:k\phantom{\rule{4.pt}{0ex}}\text{appears}\phantom{\rule{4.pt}{0ex}}\text{between}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}j\phantom{\rule{4.pt}{0ex}}(\text{inclusive})\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{4.pt}{0ex}}\text{linear}\phantom{\rule{4.pt}{0ex}}\text{extension}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}P\}$;
- (iii)
- $I=\{k\in V:k\phantom{\rule{4.pt}{0ex}}\text{lies}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{directed}\phantom{\rule{4.pt}{0ex}}\text{path}\phantom{\rule{4.pt}{0ex}}\text{from}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}j\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\omega \}$.

**Example 5.9.**

**Definition 5.10.**

**Remark 5.11.**

**Corollary 5.12.**

**Proposition 5.13.**

**Proof.**

**Proposition 5.14.**

**Proof.**

**Proposition 5.15.**

**Proof.**

#### 5.5. Toric Antichains

- combinatorially by the condition that no pair $\{i,j\}\subseteq A$ with $i\ne j$ are comparable, that is, they lie on no common chain of P, or
- geometrically by the equivalent condition that the $\left(\right|V|-|A|+1)$-dimensional subspace ${D}_{{\pi}_{A}}$ intersects the open polyhedral cone $c\left(P\right)$ in ${\mathbb{R}}^{V}$.

**Definition 5.16.**Given a toric poset P on V, say that $A\subseteq V$ is a geometric toric antichain if ${D}_{{\pi}_{A}}^{tor}$ intersects the open toric chamber $c\left(P\right)$ in ${\mathbb{R}}^{V}/{\mathbb{Z}}^{V}$.

**Proposition 5.17.**

## 6. Morphisms of Toric Posets

- ■
- quotients that correspond to projecting the toric chamber onto a flat of ${\mathcal{A}}_{tor}\left({G}_{\pi}^{\prime}\right)$ for some closed toric face partition $\pi ={cl}_{P}^{tor}\left(\pi \right)$;
- ■
- inclusions that correspond to embedding a toric chamber into a higher-dimensional chamber;
- ■
- extensions that add relations (toric hyperplanes).

#### 6.1. Quotient

**Lemma 6.1.**

**Corollary 6.2.**

**Corollary 6.3.**

#### 6.2. Inclusion

#### 6.3. Extension

- ■
- $i{\le}_{P}j$ implies $i{\le}_{{P}^{\prime}}j$;
- ■
- ${\widehat{G}}^{Hasse}\left(P\right)\subseteq {\widehat{G}}^{Hasse}\left({P}^{\prime}\right)$, where ⊆ is inclusion of edge sets;
- ■
- $c\left({P}^{\prime}\right)\subseteq c\left(P\right)$.

#### 6.4. Summary

- (i)
- quotient: Collapsing G by a partition π that preserves acyclicity of some ${\omega}^{\prime}\in \left[\omega \right]$ (projecting to a flat ${D}_{\pi}^{tor}$ of $\mathcal{A}\left({G}_{\pi}^{\prime}\right)$ for some partition $\pi ={cl}_{P}^{tor}\left(\pi \right)$).
- (ii)
- inclusion: Adding vertices (adding dimensions).
- (iii)
- extension: Adding relations (cutting the chamber with toric hyperplanes).

## 7. Toric Order Ideals and Filters

- ■
- ${x}_{{i}_{k}}={x}_{{i}_{\ell}}$ for all ${i}_{k},{i}_{\ell}$ in I;
- ■
- ${x}_{{j}_{k}}={x}_{{j}_{\ell}}$ for all ${j}_{k},{j}_{\ell}$ in J;
- ■
- ${x}_{i}\le {x}_{j}$ for all $i\in I$ and $j\in J$.

**Definition 7.1.**

- ■
- ${x}_{{i}_{k}}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1={x}_{{i}_{\ell}}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1$ for all ${i}_{k},{i}_{\ell}$ in I;
- ■
- ${x}_{{j}_{k}}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1={x}_{{j}_{\ell}}\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1$ for all ${j}_{k},{j}_{\ell}$ in J.

**Remark 7.2.**

**Proposition 7.3.**

- (i)
- I is a toric filter of $P(G,[\omega \left]\right)$;
- (ii)
- I is an ideal of $P(G,{\omega}^{\prime})$ for some ${\omega}^{\prime}\in \left[\omega \right]$;
- (iii)
- I is a filter of $P(G,{\omega}^{\prime \prime})$ for some ${\omega}^{\prime \prime}\in \left[\omega \right]$;
- (iv)
- In at least one total toric extension of $P(G,[\omega \left]\right)$, the elements in I appear in consecutive cyclic order.

**Proof.**

**Proposition 7.4.**

**Corollary 7.5.**

**Proposition 7.6.**

**Proof.**

**Example 7.7.**

**Example 7.8.**

## 8. Application to Coxeter Groups

## 9. Concluding Remarks

## Conflicts of Interest

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**Figure 2.**The hyperplane arrangement $\mathcal{A}\left(G\right)$ for $G={K}_{3}$. Three orientations in $Acyc\left(G\right)$ are shown, along with the corresponding chambers of $\mathcal{A}\left(G\right)$, and the preposet that results when contracting ${P}_{i}=P(G,{\omega}_{i})$ by the partition $\pi =\{{B}_{1}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\{1\},{B}_{2}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\{2,3\}\}$ of V. The intersection of each (closed) chamber $\overline{c\left({P}_{i}\right)}$ with ${[0,1]}^{3}$ is the order polytope $\mathcal{O}\left({P}_{i}\right)$ of ${P}_{i}$. The point y is supposed to lie on the hyperplane ${x}_{1}={x}_{2}$.

**Figure 3.**Despite the equality $P({C}_{4},\left[\omega \right])=P({K}_{4},\left[{\omega}^{\prime}\right])$, the set $I=\{1,3\}$ lies on a toric directed path $1{\to}_{tor}3$ in ${\omega}^{\prime}$ but not for any representative of $\left[\omega \right]$.

**Figure 4.**The four torically equivalent orientations to $\omega \in Acyc\left({C}_{4}\right)$, shown at left. The edges implied by toric transitivity are dashed.

**Figure 5.**The toric filters of the toric poset $P({C}_{4},\left[\omega \right])$ form a poset that is not a lattice. The vertices should be thought of as subsets of $\{1,2,3,4\}$; order does not matter. That is, 134 represents $\{1,3,4\}$.

**Figure 6.**The six torically equivalent orientations to ${\omega}^{\prime}\in Acyc\left({C}_{4}\right)$, shown at left.

**Figure 7.**The toric filters of the toric poset $P({C}_{4},\left[{\omega}^{\prime}\right])$ form a poset that happens to be a lattice. Each toric filter appears as a consecutive sequence in one of the total toric extensions, shown at right.

**Figure 8.**Two non-torically equivalent orientations $\omega \not\equiv {\omega}^{\prime}$ in $Acyc\left({C}_{5}\right)$ for which $P({C}_{5},\left[\omega \right])$ and $P({C}_{5},\left[{\omega}^{\prime}\right])$ have the same set of toric chains.

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Macauley, M.
Morphisms and Order Ideals of Toric Posets. *Mathematics* **2016**, *4*, 39.
https://doi.org/10.3390/math4020039

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Macauley M.
Morphisms and Order Ideals of Toric Posets. *Mathematics*. 2016; 4(2):39.
https://doi.org/10.3390/math4020039

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Macauley, Matthew.
2016. "Morphisms and Order Ideals of Toric Posets" *Mathematics* 4, no. 2: 39.
https://doi.org/10.3390/math4020039