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 x0, ..., xn-1, y0, ..., yn-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 xi associated with members of I and the yi 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 tI 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
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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
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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
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