A note on Janowitz's hulls of generalized orthomodular lattices (Q2706528)

A structure \((G,\vee,\wedge,(^{P(a)};a\in G),0)\) is called a generalized orthomodular lattice (GOML) if \((G,\vee,\wedge,0)\) is a lattice with smallest element \(0\), for each \(a\in G\), \(([0,a],\vee,\wedge,^{P(a)},0,a)\) is an orthomodular lattice (OML) and \(a^{P(b)}=a^{P(c)}\wedge b\) for all \(a,b,c\in G\) with \(a\leq b\leq c\). Let \({\mathcal G}=(G,\vee,\wedge,(^{P(a)};a\in G),0)\) be a GOML. For \(a,b\in G\), \(a\perp b\) is defined by \(a\leq b^{P(a\vee b)}\) and for \(M\subseteq G\), \(M^{\perp}\) is defined as \(\{x\in G |x\perp m\text{ for all }m\in M\}\). \(\{[0,x] |x\in G\}\cup\{[0,x]^{\perp} |x\in G\}\) is called the Janowitz hull of \({\mathcal G}\). It is a sublattice of the ideal lattice of \((G,\vee,\wedge)\) and an OML. An ideal \(I\) of a lattice \((L,\vee,\wedge)\) is called a prime ideal if for all \(a,b\in L\), \(a\wedge b\in L\) implies \(\{a,b\}\cap I\neq\emptyset\). Besides other results it is proved that an OML \({\mathcal L}\) is isomorphic to the Janowitz hull of a strict GOML if and only if \({\mathcal L}\) has a proper nonprincipal prime ideal.