H = hibiRing P
H = hibiRing(P, Strategy => "kernel")
H = hibiRing(P, Strategy => "4ti2")
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 $P$.
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The Hibi ring of the $n$ chain is just a polynomial ring in $n+1$ variables.
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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.
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The object hibiRing is a method function with options.