posets 1.0.0

A package for working with finite partial orders

This module provides a class Poset that encodes a finite partially ordered set (poset). Most notably, this module can efficiently compute flag vectors, the ab-index and the cd-index. Quasigraded posets, in the sense of [2], can be encoded and the ab-index and cd-index of quasigraded posets can be computed. Latex code for Hasse diagrams can be produced with a very flexible interface. There are methods for common operations and constructions such as Cartesian products, disjoint unions, interval lattices, lattice of ideals, etc. Various examples of posets are provided such as Boolean algebras, the face lattice of the $n$-dimensional cube, (noncrossing) partition lattices, the type $A_n$ Bruhat and weak orders, uncrossing orders etc. General subposets can be selected as well as particular ones of interest such as intervals and rank selections. Posets from this module can also be converted to and from posets from sagemath and Macaulay2.

Terminology and notation on posets generally follows [3] and [1].

Installation

Download the whl file here and install it with pip via python -m pip posets-*-py3-none-any.whl.

Building

Building the package requires hatch to be installed. Running make will build the package with a timestamp in the version, to build without the timestamp in the version run make RELEASE=0 (the value of 0 is arbitrary, you just need to set RELEASE to any nonemtpy value). This will make a whl file at dist/posets-<version>-py3-none-any.whl that you can install with pip.

The documentation can be built in pdf form by running make docs from the base directory, this will also use a timestamp in the version unless you use RELEASE=0. Compilation requires LaTeXto be installed with the packages pgf/tikz, graphicx, fancyvrb, amsmath, amsssymb, scrextend, mdframed and hyperref as well as the python module pydox. pydox can be obtained from github.com/WilliamGustafson/pydox.git. You must either place the script pydox.py somewhere in your path named pydox or call make with make PYDOX=[path to pydox] docs.

Testing

You can run the tests via make test, this requires pytest to be installed. You can create an html coverage report, output to tests/htmlcov/index.html, with make coverage. Making the coverage report requires pytest, coverage and the pytest-cov plugin. Note, coverage on hasseDiagram.py is very low because testing drawing functions cannot be easily automated.

Publishing

After building the package it can be published to PyPi by running make publish RELEASE=0, running make publish will publish to TestPyPi. This requires twine and gpg to be installed. Additionally, the publishing command expects there to be an encrypted api token named pypi.token.gpg (or test.pypi.token.gpg for TestPyPi) at the project root. This will also build the package if it is not built already. If you have an unencrypted api token from PyPi named pypi.token you can encrypt it via:

gpg --encrypt --symmetric --output pypi.token.gpg pypi.token

Usage

Here we give an introduction to using the posets module.

In the code snippets below we assume the module is imported via

from posets import *

Constructing a poset:

P = Poset(relations={'':['a','b'],'a':['ab'],'b':['ab']})
Q = Poset(relations=[['','a','b'],['a','ab'],['b','ab']])
R = Poset(elements=['ab','a','b',''], less=lambda x,y: return x in y)
S = Poset(zeta = [[0,1,1,1],[0,0,0,1],[0,0,0,1],[0,0,0,0]], elements=['','a','b','ab'])

Built in examples (see page ):

Boolean(3) #Boolean algebra of rank 3
Cube(3) #Face lattice of the 3-dimensional cube
Bruhat(3) #Bruhat order on symmetric group of order 3!
Bnq(n=3,q=2) #Lattice of subspaces of F_2^3
DistributiveLattice(P) #lattice of ideals of P
Intervals(P) #meet semilattice of intervals of P

These examples come with default drawing methods, for example, when making latex code by calling DistributiveLattice(P).latex() the resulting figure depicts elements of the lattice as Hasse diagrams of $P$ with elements of the ideal highlighted (again, see page ). Note, you will have to set the height, width and possibly nodescale parameters in order to get sensible output.

Two posets compare equal when they have the same set of elements and the same zeta values (i.e. the same order relation with the same weights):

P == Q and Q == R and R == S #True
P == Poset(relations={'':['a','b']}) #False
P == Poset(relations={'':['ab'],'a':['ab'],'b':['ab']}) #False
P == Poset(zeta=[[0,1,1,2],[0,0,0,3],[0,0,0,4],[0,0,0,0]],
    elements=['','a','b','ab']) #False

Use is_isomorphic or PosetIsoClass to check whether posets are isomorphic:

P.is_isomorphic(Boolean(2)) #True
P.isoClass()==Boolean(2).isoClass() #True
P.is_isomorphic(Poset(relations={'':['a','b']})) #False

Viewing and creating Hasse diagrams:

P.show() #displays a Hasse diagram in a new window
P.latex() #returns latex code: \begin{tikzpicture}...
P.latex(standalone=True) #latex code for a
#standalone document: \documentclass{preview}...
display(P.img()) #Display a poset when in a Jupyter notebook
#this uses the output of latex()

Computing invariants:

Cube(2).fVector() #{(): 1, (1,): 4, (2,): 4, (1, 2): 8}
Cube(2).hVector() #{(): 1, (1,): 3, (2,): 3, (1, 2): 1}
Boolean(5).sparseKVector() #{(3,): 8, (2,): 8, (1, 3): 4, (1,): 3, (): 1}
Boolean(5).cdIndex() #Polynomial({'ccd': 3, 'cdc': 5, 'dd': 4, 'dcc': 3, 'cccc': 1})
print(Boolean(5).cdIndex()) #c^{4}+3c^{2}d+5cdc+3dc^{2}+4d^{2}

Polynomial operations:

#Create noncommutative polynomials from dictionaries,
#keys are monomials, values are coefficients
p=Polynomial({'ab':1})
q=Polynomial({'a':1,'b':1})

#get and set coefficients like a dictionary
q['a'] #1
q['x'] #0
p['ba'] = 1

#print latex
str(p) #ab+ba

#basic arithmetic, polynomials form a real algebra
p+q #ab+ba+a+b
p*q #aba+ab^{2}+ba^{2}+bab
q*p #a^{2}b+aba+bab+b^{2}a
2*p #2ab+2ba
p**2 #abab+ab^{2}a+ba^{2}b+baba
p**(-1) #raises NotImplementedError
p**q #raises NotImplementedError

#substitutions and conversions
p.sub(q,'a') #ab+ba+2b^{2} substitute q for a in p
p.abToCd() #d rewrite a's and b's
#in terms of c=a+b and d=ab+ba when possible
Polynomial({'c':1,'d':1}).cdToAb() #a+b+ab+ba rewrite c's and d's
#in terms of a's and b's

Converting posets to and from SageMath:

P.toSage() #Returns a SageMath class, must be run under sage
Poset.fromSage(Q) #Take a poset Q made with SageMath and return an instance of Poset

Converting to and from Macaulay2:

-- In M2
load "convertPosets.m2" --Also loads Python and Posets packages
import "posets" --This module must be installed to system version of python
P = posets\@\@Boolean(3) --Calling python functions
pythonPosetToMac(P) --Returns an instance of the M2 class Posets
macPosetToPython(Q) --Take a poset made with M2 and return an
--instance of the python class Poset

Quasigraded posets:

#Provide the zeta and rank functions explicitly
#To construct a 2-chain with top two elements rank 2 and 3
#and with zeta value -1 between minimum and the element covering it:
T = Poset([[1,-1,1],[1,1],[1]], ranks=[[0],[],[1],[2]])

The poset T above is from [2, Example 6.14] with $M$ taken to be the 3-dimensional solid torus.

You can calculate the flag vectors and the cd-index just as you would for a classical poset, for example, T.cdIndex() returns the polynomial $\textup{\textbf{c}}^2-2\textup{\textbf{d}}$.

When plotting a quasigraded poset by default only the underlying poset is shown with element heights based on rank, the zeta values are not shown. If you wish to display the zeta values you can use the class ZetaHasseDiagram to draw a Hasse diagram of your poset with an element $p$ depicted as the associated filter, namely the subposet $\{q:q\ge p\}$, and with elements of the filters labeled by the corresponding zeta value. To do so, either construct the poset with hasse_class=ZetaHasseDiagram such as in Poset([[1,-1,1],[1,1],[1]], ranks=[[0],[],[1],[2]],hasse_class=ZetaHasseDiagram) or set the Hasse diagram attribute on the poset as below:

T = Poset([[1,-1,1],[1,1],[1]], ranks=[[0],[],[1],[2]])
T.hasseDiagram = ZetaHasseDiagram(T)

You can also represent elements with ideals instead of filters by passing filters=False. See ZetaHasseDiagram and SubposetsHasseDiagram for a thorough explanation of the options.

References

1. Garrett Birkhoff. 1967. Lattice theory. American Mathematical Society, Providence, R.I.

2. Richard Ehrenborg, Mark Goresky, and Margaret Readdy. 2015. Euler flag enumeration of whitney stratified spaces. Adv. Math. (N. Y.) 268: 85–128.

3. Richard P Stanley. 2012. Enumerative combinatorics. Volume 1. Cambridge University Press, Cambridge.