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, 2} {0, 1, 2} {0, 1, 2, 3} o1 = ---------------------------------------------------------- t t - t t {0, 1} {0, 2} {0} {0, 1, 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") ------------------------------------------------- 4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team. 4ti2 comes with ABSOLUTELY NO WARRANTY. This is free software, and you are welcome to redistribute it under certain conditions. For details, see the file COPYING. ------------------------------------------------- Using 64 bit integers. 4ti2 Total Time: 0.00 secs. using temporary file name /tmp/M2-13367-0/0 QQ[t , t , t , t , t , t ] {} {0} {0, 1} {0, 2} {0, 1, 2} {0, 1, 2, 3} o3 = ---------------------------------------------------------- t t - t t {0, 1} {0, 2} {0} {0, 1, 2} o3 : QuotientRing |