Note on p-ideals set of orthomodular lattices

Abstract

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.

Keywords

• orthomodular lattices
• $ om $-$ l $-algebras
• $ p $-ideals
• $ l $-ideals
• well ordered