Math.Combinatorics.Poset

Documentation

newtype Poset t

A poset is represented as a pair (set,po), where set is the underlying set of the poset, and po is the partial order relation

Constructors

Poset ([t], t -> t -> Bool)

Instances

Eq t => Eq (Poset t)

Show t => Show (Poset t)

implies :: Bool -> Bool -> Bool

isReflexive :: ([t], t -> t -> Bool) -> Bool

isAntisymmetric :: Eq a => ([a], a -> a -> Bool) -> Bool

isTransitive :: ([t], t -> t -> Bool) -> Bool

isPoset :: Eq t => ([t], t -> t -> Bool) -> Bool

poset :: Eq t => ([t], t -> t -> Bool) -> Poset t

intervals :: Poset t -> [(t, t)]

interval :: Poset t -> (t, t) -> [t]

chainN :: Int -> Poset Int

A chain is a poset in which every pair of elements is comparable (ie either x <= y or y <= x). It is therefore a linear or total order. chainN n is the poset consisting of the numbers [1..n] ordered by (<=)

antichainN :: Int -> Poset Int

An antichain is a poset in which distinct elements are incomparable. antichainN n is the poset consisting of [1..n], with x <= y only when x == y.

divides :: Integral a => a -> a -> Bool

divisors :: Integral a => a -> [a]

posetD :: Int -> Poset Int

posetD n is the lattice of (positive) divisors of n

powerset :: [t] -> [[t]]

posetB :: Int -> Poset [Int]

posetB n is the lattice of subsets of [1..n] ordered by inclusion

partitions :: [t] -> [[[t]]]

isRefinement :: Ord a => [[a]] -> [[a]] -> Bool

posetP :: Int -> Poset [[Int]]

posetP n is the lattice of set partitions of [1..n], ordered by refinement

intervalPartitions :: (Num a, Eq a) => [a] -> [[[a]]]

isInterval :: (Num a, Eq a) => [a] -> Bool

intervalPartitions2 :: [t] -> [[[t]]]

integerPartitions :: (Ord t, Num t) => t -> [[t]]

isIPRefinement :: (Ord a, Num a) => [a] -> [a] -> Bool

posetIP :: Int -> Poset [Int]

posetIP n is the poset of integer partitions of n, ordered by refinement

subspaces :: (Num a, Eq a) => [a] -> Int -> [[[a]]]

isSubspace :: (Num a, Eq a) => [[a]] -> [[a]] -> Bool

posetL :: (Eq fq, Num fq) => Int -> [fq] -> Poset [[fq]]

posetL n fq is the lattice of subspaces of the vector space Fq^n, ordered by inclusion. Subspaces are represented by their reduced row echelon form. Example usage: posetL 2 f3

subposet :: Poset a -> (a -> Bool) -> Poset a

The subposet of a poset satisfying a predicate

dsum :: Poset a -> Poset b -> Poset (Either a b)

The direct sum of two posets

dprod :: Poset a -> Poset b -> Poset (a, b)

The direct product of two posets

dual :: Poset a -> Poset a

The dual of a poset

hasseDigraph :: Eq a => Poset a -> Digraph a

Given a poset (X,<=), we say that y covers x, written x -< y, if x < y and there is no z in X with x < z < y. The Hasse digraph of a poset is the digraph whose vertices are the elements of the poset, with an edge between every pair (x,y) with x -< y. The Hasse digraph can be represented diagrammatically as a Hasse diagram, by drawing x below y whenever x -< y.

reachabilityPoset :: Ord a => Digraph a -> Poset a

Given a DAG (directed acyclic graph), return the poset consisting of the vertices of the DAG, ordered by reachability. This can be used to recover a poset from its Hasse digraph.

isOrderPreserving :: (a -> b) -> Poset a -> Poset b -> Bool

orderIsos01 :: Poset a -> Poset a1 -> [[(a, a1)]]

isOrderIso :: (Ord a, Ord b) => Poset a -> Poset b -> Bool

Are the two posets order-isomorphic?

orderIsos :: (Ord a, Ord b) => Poset a -> Poset b -> [[(a, b)]]

Find all order isomorphisms between two posets

orderAuts1 :: Ord b => Poset b -> [[(b, b)]]

isLinext :: Poset t -> [t] -> Bool

A linear extension of a poset is a linear ordering of the elements which extends the partial order. Equivalently, it is an ordering [x1..xn] of the underlying set, such that if xi <= xj then i <= j.

linexts :: Poset a -> [[a]]

Linear extensions of a poset