The Hibi ring of $P$ is a monomial algebra generated by the monomials which generate the Hibi ideal (hibiIdeal). That is, the monomials built 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 monomials 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 $I$.
This method returns the toric quotient algebra isomorphic to the Hibi ring. The ideal is the ideal of Hibi relations. The generators of the PolynomialRing $H$ is built over are of the form $t_I$ where $I$ is an order ideal of $P$.
i1 : hibiRing booleanLattice 2
QQ[t , t , t , t , t , t ]
{} {0} {0, 1} {0, 1, 2} {0, 2} {0, 1, 2, 3}
o1 = ----------------------------------------------------------
t t - t t
{0} {0, 1, 2} {0, 1} {0, 2}
o1 : QuotientRing
|
The Hibi ring of the $n$ chain is just a polynomial ring in $n+1$ variables.
i2 : hibiRing chain 4
o2 = QQ[t , t , t , t , t ]
{} {0} {0, 1} {0, 1, 2} {0, 1, 2, 3}
o2 : PolynomialRing
|
In some cases, it may be faster to use the FourTiTwo method toricGroebner to generate the Hibi relations. Using the Strategy "4ti2" tells the method to use this approach.
i3 : hibiRing(divisorPoset 6, Strategy => "4ti2")
QQ[t , t , t , t , t , t ]
{} {0} {0, 1} {0, 1, 2} {0, 2} {0, 1, 2, 3}
o3 = ----------------------------------------------------------
- t t + t t
{0} {0, 1, 2} {0, 1} {0, 2}
o3 : QuotientRing
|
The object hibiRing is a method function with options.