# Reference request: Matroid cryptomorphisms for arbitrary monomial ideals

For a matroid $$M$$ let $$C$$ be the circuit ideal of $$M$$, that is, the Stanley-Reisner ideal of independence complex of $$M$$. Then there are simple ideal-theoretic operations that take $$C$$ to the facet ideal $$F$$ of $$M$$ (and vice versa). Moreover, Alexander duality takes $$C$$ to the cofacet ideal $$F^*$$ of $$M$$. So it is possible to "idealize" the matroid cryptomorphisms between (co)circuits to (co)facets.

Now drop the assumption on $$C$$ that it is square-free and its generators satisfy the circuit exchange axiom. All of the operations needed above are defined for the pair $$(\tau, C)$$ where $$\tau$$ is a monomial that is divisible by the least common multiple of the minimal generators of $$C$$. So for such a pair it is not hard to construct a tuple $$(C, F, F^*, C^*)$$ of monomial ideals that satisfy the same "idealized matroid cryptomorphisms" as matroids do. In particular, for $$F$$ appropriated defined we have $$F^* = \{\tau/f | f \in G(F)\} = C^{[\mathbf{t}]}$$ where $$\tau = \mathbf{x}^{\mathbf{t}}$$.

Question: Have such tuples of monomial ideals been explicitly considered in the literature?

Here are some examples where some of these ideals appear.

• For a simplicial complex with Stanley-Reisner ideal $$C$$ and facet ideal $$F$$ the triples $$(C, F, F^* := C^{[\boldsymbol{1}]})$$ are considered in this paper (see Lemma 1.2).
• If $$F$$ is a polymatroidal ideal minimally generated by $$G$$ and $$\tau$$ is any monomial divisible by $$\mathrm{lcm} (G)$$, then it's easy to show that $$F^* = \langle \tau/f | f \in G \rangle$$ is a polymatroidal ideal. Such ideals are natural generalizations of matroid facet ideals. While they have been widely studied, I have found no reference in the literature to a similar generalization of matroid circuit ideals.
• Riemann-Roch ideals are those artinian monomial ideals $$C$$ for which there exists a $$\mathbf{x}^{\mathbf{t}}$$ such that $$F^* = C^{[\mathbf{t}]} = \{\tau/f^* | f^* \in G(F^*)\}$$. So they are similar in flavor to (identically) self-dual matroids.