The chain algebra of a pure poset
Dancheng Lu
TL;DR
This work extends the chain algebra construction from finite distributive lattices to finite pure posets by defining $K[C_P]$ as the $K$-algebra generated by maximal-chain monomials and studying the associated chain semigroup $C_P$. It establishes a combinatorial description of $C_P$ via $f$-parallelizations and the Marriage Lemma, proving normality and providing a Krull-dimension formula that generalizes known results for lattices. The authors connect $K[C_P]$ to the chain polytope $D_P$, show $D_P$ has the integer decomposition property, and characterize indecomposability in terms of ordinal sums and complete bipartite graphs. They determine the canonical module $\omega_{K[C_P]}$ via an ideal $K_P$ in terms of linear inequalities, and compute the dimension and $a$-invariant to relate to regularity and Gorenstein properties. For width-2 posets, they classify when the chain algebra is Gorenstein or nearly Gorenstein, using a decomposition into basic posets and antichains, yielding precise conditions on the induced type data.
Abstract
We extend the notion of chain algebra, originally defined in \cite{GN} for finite distributive lattices, to that of finite pure posets. We show this algebra corresponds to the Ehrhart ring of a (0,1)-polytope, termed the chain polytope, and characterize the indecomposability of this polytope. Furthermore, we prove the normality of the chain algebra, describe its canonical module, and extend one of main results from \cite{GN} by computing its Krull dimension. For width-2 pure posets, we determine the algebra's regularity and conditions for it to be Gorenstein or nearly Gorenstein.