#!/usr/bin/env python
# coding: utf-8

# # Kappa map for lattices
# 
# This is a manual of [kappa.py](https://github.com/haruhisa-enomoto/kappa-map-for-lattices/blob/main/kappa.py), an additional module for [SageMath](https://www.sagemath.org/).
# 
# AUTHOR:
# [Haruhisa Enomoto](http://haruhisa-enomoto.github.io/)
# 
# REFERENCES:
# 
# - [BCZ] E. Barnard, G. Todorov, S. Zhu,
#   Dynamical combinatorics and torsion classes,
#   J. Pure Appl. Algebra 225 (2021), no. 9, 106642.
# - [RST] N. Reading, D. E. Speyer, H. Thomas,
#   The fundamental theorem of finite semidistributive lattices,
#   arXiv:1907.08050.

# A function `add_kappa()` in  [kappa.py](https://github.com/haruhisa-enomoto/kappa-map-for-lattices/blob/main/kappa.py) adds to a Sage class [`sage.combinat.posets.lattices.FiniteLatticePoset`](https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/lattices.html#sage.combinat.posets.lattices.FiniteLatticePoset) class methods
# - `kappa`
# - `extended_kappa`
# - `kappa_dual`
# - `extended_kappa_dual`
# 
# To use it, first import `add_kappa` function from `kappa.py` and execute `add_kappa()`.

# In[1]:


from kappa import add_kappa
add_kappa()


# By this, we can use the above methods for an instance of a Sage class [`FiniteLatticePoset`](https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/lattices.html#sage.combinat.posets.lattices.FiniteLatticePoset).

# ## Kappa map
# 
# Let $L$ be a finite lattice. An element $j$ of $L$ is *join-irreducible*
# if $j = \bigvee A$ for a subset $A$ of $L$ implies $j \in A$.
# This is equivalent to that there is a unique element $j_*$ which is covered by $j$.
# Dually, we define *meet-irreducible* elements.
# 
# A *kappa map $\kappa(j)$* for a join-irreducible element $j$ of $L$ is defined, if exists, as a *maximum* element
# $x$ satisfying $j \wedge x = j_*$. Then $\kappa(j)$ is meet-irreducible if exists
# (see e.g. [RST, Theorem 2.28]).

# ### kappa(j, check = True)
# Return $\kappa(j)$ for a join-irreducible element $j$
# of a finite lattice ``self`` if it exists.
# 
# INPUT:
#  - ``j`` -- an element of ``self``,
#       which is expected to be join-irreducible
# 
#  - ``check`` -- a Boolean (default: `True`),
#       whether to check that ``j`` is indeed join-irreducible
# 
# OUTPUT:
# 
# an element of ``self``, or `None` if it does not exist.
# 
# #### See also
# A method [`sage.combinat.posets.hasse_diagram.HasseDiagram.kappa`](https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/hasse_diagram.html#sage.combinat.posets.hasse_diagram.HasseDiagram.kappa)
# 
# EXAMPLE:

# In[2]:


L = posets.PentagonPoset()
L.plot(figsize = 3)


# In[3]:


L.kappa(1)


# In[4]:


L.kappa(4)


# In[5]:


L_new = LatticePoset({1: [2, 3], 2: [4, 5], 3: [6], 4: [6], 5: [6]})
L_new.plot(figsize = 4)


# In[6]:


L_new.kappa(2)


# In[7]:


L_new.kappa(3) is None


# ## Dual kappa map
# 
# Let $m$ be a meet-irreducible element in a finite lattice $L$, and $m_*$ a unique element covering $m$.
# A *dual kappa map $\kappa^d(m)$* for a meet-irreducible element $j$ of $L$ is defined, if exists, as a *minimum* element
# $x$ satisfying $x \vee m = m_*$. Then $\kappa^d(m)$ is join-irreducible if exists.
# Moreover, $\kappa$ and $\kappa^d$ are mutually inverse bijecionts between the set of join-irreducibles and meet-irreducibles if and only if
# $L$ is semidistributive, see e.g. [RST Corollary 2.29].

# ### kappa_dual(m, check = True)
# Return $\kappa^d(m)$ for a meet-irreducible element $m$
# in a finite lattice if it exists.
# 
# INPUT:
#  - ``m`` -- an element of ``self``,
#       which is expected to be meet-irreducible
# 
#  - ``check`` -- a Boolean (default: `True`),
#       whether to check that ``m`` is indeed join-irreducible
# 
# OUTPUT:
# 
# an element of ``self``, or `None` if it does not exist.
# 
# #### See also
# A method [`sage.combinat.posets.hasse_diagram.HasseDiagram.kappa_dual`](https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/hasse_diagram.html#sage.combinat.posets.hasse_diagram.HasseDiagram.kappa_dual)
# 
# EXAMPLE:

# In[8]:


L = posets.PentagonPoset()
L.kappa_dual(3)


# ## Extended (dual) kappa map
# 
# An *extended kappa map $\overline{\kappa}(x)$* for an element $x$ of a finite lattice $L$ is defined as follows. First, let $x = j_1 \vee \cdots \vee j_n$ be a canonical join representation of $x$ if exists. Then $\overline{\kappa}(x) := \kappa(j_1) \wedge \cdots \wedge \kappa(j_n)$ if $\kappa(j_1), \dots, \kappa(j_n)$ are defined. This map is well-defined if $L$ is semidistributive.
# 
# An *extended dual kappa map* $\overline{\kappa^d}(x)$ is defined completely dually. This map is well-defined and is an inverse of $\overline{\kappa}$ if $L$ is semidistributive.

# ### extended_kappa(x)
# Return $\overline{\kappa}(x)$ for an element $x$
# of a finite lattice ``self`` if it exists.
# 
# INPUT:
# 
# - ``x`` -- an element of ``self``
# 
# OUTPUT:
# 
# an element of ``self``, or `None` if it does not exist.
# 
# REFERENCE:
# 
# - [BCZ Definition 4.4.1]

# In[9]:


L = posets.PentagonPoset()
L.plot(figsize = 4)


# In[10]:


for x in L:
    print(f"extended_kappa({x}) is {L.extended_kappa(x)}")


# ### extended_kappa_dual(x)
# Return $\overline{\kappa^d}(x)$ for an element $x$
# in a finite lattice ``self`` if it exists.
# 
# INPUT:
# 
# - ``x`` -- an element of ``self``
# 
# OUTPUT:
# 
# an element of ``self``, or `None` if it does not exist.
# 
# REFERENCE:
# 
# - [BCZ Definition 4.4.1]
# 
# EXAMPLE:

# In[11]:


L = posets.PentagonPoset()
for x in L:
    print(f"extended_kappa_dual({x}) is {L.extended_kappa_dual(x)}")