On incidence Hopf algebras for non-hereditary families

Zaf hinted at this in class today:

When a family F of posets is not hereditary, one can “force it to be hereditary” by enlarging it, considering the (bigger) family F’={products of intervals of posets in F}. This new family F’ is hereditary. If F has an equivalence relation \sim which satisfies that

(P \sim Q) => there is a bijection f:P \rightarrow Q such that [0,p] \sim [0, f(p)] and [p,1] \sim [f(p),1] for all p in P

then that gives an equivalence relation on F’ (how?) so that the result is indeed a reduced congruence.

I mentioned this in class explicitly in the case when F is closed under taking subintervals, and when P, Q in F imply that PxQ is not in F. This is a special case of the above, and in this case we fully understand the multiplicative structure of the incidence Hopf algebra (which is free commutative).

I also mentioned this implicitly in the case of graphs.

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