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].

The full documentation for the current version can be found here.

Installation

Install with pip via python -m pip install posets. Alternatively, download the whl file here and install it with pip via python -m pip posets-*-py3-none-any.whl.

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 an 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 TypeError
p**q #raises TypeError

#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:
G2 = Poset([1,2,1, 1,1, 1],ranks=[[0],[],[],[1],[2]], flat_zeta=True)

The poset G2 above is from [2, Example 6.14] with $M$ taken to be the manifold whose boundary is the genus 2 surface.

You can calculate the flag vectors and the cd-index just as you would for a classical poset, for example, G2.cdIndex() returns the polynomial $\textbf{c}^3-2\textbf{d}\textbf{c}$.

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([1,2,1, 1,1, 1],ranks=[[0],[],[],[1],[2]], flat_zeta=True or set the Hasse diagram attribute on the poset as below:

G2 = Poset([1,2,1, 1,1, 1],ranks=[[0],[],[],[1],[2]], flat_zeta=True)
G2.hasseDiagram = ZetaHasseDiagram(G2)

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.