Math.Combinatorics.FiniteGeometry

Constructions of the finite geometries AG(n,Fq) and PG(n,Fq), their points, lines and flats, together with the incidence graphs between points and lines.

Documentation

ptsAG :: Int -> [a] -> [[a]]

ptsAG n fq returns the points of the affine geometry AG(n,Fq), where fq are the elements of Fq

ptsPG :: Num a => Int -> [a] -> [[a]]

ptsPG n fq returns the points of the projective geometry PG(n,Fq), where fq are the elements of Fq

closureAG :: (Ord a, FiniteField a) => [[a]] -> [[a]]

Given a list of points in AG(n,Fq), return their closure, the smallest flat containing them

closurePG :: (Ord a, FiniteField a) => [[a]] -> [[a]]

Given a set of points in PG(n,Fq), return their closure, the smallest flat containing them

data ZeroOneStar

Constructors

Zero

One

Star

Instances

Eq ZeroOneStar

Show ZeroOneStar

flatsPG :: Num a => Int -> [a] -> Int -> [[[a]]]

flatsPG n fq k returns the k-flats in PG(n,Fq), where fq are the elements of Fq. The returned flats are represented as matrices in reduced row echelon form, the rows of which are the points that generate the flat. The full set of points in the flat can be recovered by calling closurePG

flatsAG :: Num a => Int -> [a] -> Int -> [[[a]]]

flatsAG n fq k returns the k-flats in AG(n,Fq), where fq are the elements of Fq.

linesPG :: Num a => Int -> [a] -> [[[a]]]

The lines (1-flats) in PG(n,fq)

linesAG :: Num a => Int -> [a] -> [[[a]]]

The lines (1-flats) in AG(n,fq)

incidenceGraphPG :: (Ord a, FiniteField a) => Int -> [a] -> Graph (Either [a] [[a]])

Incidence graph of PG(n,fq), considered as an incidence structure between points and lines

incidenceGraphAG :: (Ord a, FiniteField a) => Int -> [a] -> Graph (Either [a] [[a]])

Incidence graph of AG(n,fq), considered as an incidence structure between points and lines