# hibiIdeal -- produces the Hibi ideal of a poset

## Synopsis

• Usage:
H = hibiIdeal P
• Inputs:
• P, an instance of the type Poset,
• Optional inputs:
• CoefficientRing => a ring, default value QQ, which specifies the coefficient ring of the PolynomialRing $H$ is constructed in
• Outputs:
• H, , the Hibi ideal of $P$

## Description

The Hibi ideal of $P$ is a MonomialIdeal built over a ring in $2n$ variables $x_0, \ldots, x_{n-1}, y_0, \ldots, y_{n-1}$, where $n$ is the size of the ground set of $P$. The generators of the ideal are in bijection with order ideals in $P$. Let $I$ be an order ideal of $P$. Then the associated monomial is the product of the $x_i$ associated with members of $I$ and the $y_i$ associated with non-members of $I$.

 i1 : hibiIdeal chain 3 o1 = monomialIdeal (x x x , x x y , x y y , y y y ) 0 1 2 0 1 2 0 1 2 0 1 2 o1 : MonomialIdeal of QQ[x ..x , y ..y ] 0 2 0 2

• hibiRing -- produces the Hibi ring of a poset
• orderIdeal -- computes the elements below given elements in a poset
• principalOrderIdeal -- computes the elements below a given element in a poset

## Ways to use hibiIdeal :

• "hibiIdeal(Poset)"

## For the programmer

The object hibiIdeal is .