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For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<p_n$. I …

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, $b …

A well known property of finite modular lattices is that they have the same number of join-irreducible and meet-irreducible elements. I was wondering if there exists a natural bijection between these …

I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently …

I think I might've stumbled upon the answer I was looking for. It's the correspondence between interval greedoids and semimodular lattices. An interval greedoid is a pair $(E,\mathcal F)$ where $\math …

This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says tha …

I learned that there is at least one precise answer to this exact question (more precise than my previous answer). The bijection between finite posets and finite distributive lattices and the bijectio …