Note on $p$-ideals set of orthomodular lattices

This paper mainly discusses the problems raised in Kalmbach's book: When are the $p$-ideals of an irreducible orthomodular lattice well ordered under set inclusion? We give three classes of orthomodular lattices whose $p$-ideals set is a chain under set inclusion. Furthermore, this article also provides a sufficient and necessary condition for the $p$-ideals set of orthomodular lattices to be a chain under set inclusion from the perspective of $L$-algebras, which gives a new point to solve the questions. Moveover, we also give some characterizations about central elements of orthomodular lattice.