map(Poset,Poset,List) -- make a poset map

Function: map
Usage:

map(Q, P, L)
Inputs:
- P, an instance of the type Poset, the source poset
- Q, an instance of the type Poset, the target poset
- L, a list, of length #(vertices P) and whose $i$th entry specifies the image of the $i$th element of P, or a List of Options whose keys are vertices P and whose values are in vertices Q, or a HashTable whose keys are vertices P and whose values are in vertices Q
Optional inputs:
- Degree => ..., default value null,
- DegreeLift => ..., default value null,
- DegreeMap => ..., default value null,
Outputs:
- a map of posets, the poset map from P to Q specified by L.

Description

A poset map is a monotone function $f: P\rightarrow Q$ from a poset $P$ to a poset $Q$, so $f(p)\leq f(q)$ whenever $p\leq q$.

Let $P = [2]$ and $Q = [3] \times [3]$. Here are the three ways to obtain the map $\phi: P \rightarrow Q$ given by $\phi(1) = (1,1)$ and $\phi(2) = (1,2)$.

i1 : P = chain 2

o1 = P

o1 : Poset

i2 : Q = product(chain 3, chain 3)

o2 = Q

o2 : Poset

i3 : f = map(Q, P, {{1,1}, {1,2}})

o3 = PosetMap{1 => {1, 1}}
              2 => {1, 2}
              source => P
              target => Q

o3 : PosetMap

i4 : g = map(Q, P, {1=>{1,1}, 2=>{1,2}})

o4 = PosetMap{1 => {1, 1}}
              2 => {1, 2}
              source => P
              target => Q

o4 : PosetMap

i5 : h = map(Q, P, hashTable {1=>{1,1}, 2=>{1,2}})

o5 = PosetMap{1 => {1, 1}}
              2 => {1, 2}
              source => P
              target => Q

o5 : PosetMap

i6 : f == g

o6 = true

i7 : g == h

o7 = true

Potentially, some of the vertices of the target may be options. When there is ambiguity, the list elements will be taken as vertices of the target as in the first of the three ways above.

i8 : P = poset({1}, {})

o8 = P

o8 : Poset

i9 : Q = poset({1, 1=>1}, {{1, 1=>1}})

o9 = Q

o9 : Poset

i10 : f = map(Q, P, {1=>1})

o10 = PosetMap{1 => 1 => 1}
               source => P
               target => Q

o10 : PosetMap

i11 : f 1

o11 = 1 => 1

o11 : Option

Ways to use this method:

map(Poset,Poset,HashTable)
map(Poset,Poset,List) -- make a poset map

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MacaulayPosets.m2:1624:0.