Recall that a $P$-partition for a naturally labeled poset $P$ on vertices $1, \ldots, n$ is a function $f: P \rightarrow \mathbb{NN}$ which is order-reversing, i.e., if $i < j$ in $P$ then $f(i) \geq f(j)$ in $\mathbb{NN}$. To a $P$-partition $f$ we can assign the monomial $t_1^{f(1)} \ldots t_n^{f(n)}$. The $P$-partition ring is the ring spanned by the monomials corresponding to $P$-partitions.
The $P$-partition ring is more simply generated by the monomials corresponding to the connected order ideals of $P$. This method returns the toric quotient algebra, whose toric ideal is minimally generated, isomorphic to the $P$-partition ring.
i1 : P = poset {{1,2}, {2,4}, {3,4}, {3,5}};
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i2 : pPartitionRing P
QQ[t , t , t , t , t , t ]
{3} {3, 4} {0} {0, 1} {0, 1, 2, 3} {0, 1, 2, 3, 4}
o2 = -----------------------------------------------------------------
t t - t t
{3, 4} {0, 1, 2, 3} {3} {0, 1, 2, 3, 4}
o2 : QuotientRing
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In some cases, it may be faster to use the FourTiTwo method toricGroebner to generate the toric relations. Using the Strategy "4ti2" tells the method to use this approach.
i3 : pPartitionRing(divisorPoset 6, Strategy => "4ti2")
QQ[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
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The object pPartitionRing is a method function with options.