**151. a.** [2+] Let *P* be a *p*-element poset, and let *S* ⊆ [*p* − 1] such that *β*_{J(P)}(*S*) 0. Show that if *T* ⊆ *S*, then *βJ* (*P*)(*T*) 0.

**b.** [5–] Find a “nice” characterization of the collections Δ of subsets of [*p* − 1] for which there exists a *p*-element poset *P* satisfying

**c.** [2+] Show that (a) continues to hold if we replace *J*(*P*) with any finite supersolvable lattice *L* of rank *p*.

**152. a.** [2+] Let *P* be a finite naturally labeled poset. Construct explicitly ...

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