Forcing for hL and hd (Q2717737)

In this paper the authors consider two cardinal functions for Boolean algebras, the hereditary density hd and the hereditary Lindelöf degree hL. They investigate the attainment behavior of these two functions as well as of some related functions. This answers some questions of Monk. They show consistency of different attainment behaviour and show that this is the best what can be expected.NEWLINENEWLINENEWLINETo obtain their results, they show that consistently there is a Boolean algebra \({\mathcal B}\) of size \(\lambda\) in which there is a strictly increasing \(\lambda\)-sequence of ideals but every ideal in \({\mathcal B}\) is generated by less than \(\lambda\) elements. They also show that consistently there is a Boolean algebra \({\mathcal B}\) of size \(\lambda\) in which there is a strictly decreasing \(\lambda\)-sequence of ideals but every homomorphic image of \({\mathcal B}\) has algebraic density less than \(\lambda\).