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.
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 |
i2 : hibiRing chain 4 o2 = QQ[t , t , t , t , t ] {} {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} o2 : PolynomialRing |
i3 : hibiRing(divisorPoset 6, Strategy => "4ti2") using temporary file name /tmp/M2-26711-0/0 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 |