---
abstract: |
    We show that, consistently, for some regular cardinals $\theta<\lambda$, there exist a Boolean algebra ${{\mathbb B}}$ such that $|{{\mathbb B}}|=\lambda^+$ and for every subalgebra ${{\mathbb B}}'\subseteq{{\mathbb B}}$ of size $\lambda^+$ we have ${{\rm Depth}}({{\mathbb B}}')=
    \theta$.
address:
- |
    Department of Mathematics\
    University of Nebraska at Omaha\
    Omaha, NE 68182-0243, USA\
    and Mathematical Institute of Wroclaw University\
    50384 Wroclaw, Poland
- |
    Institute of Mathematics\
    The Hebrew University of Jerusalem\
    91904 Jerusalem, Israel\
    and Department of Mathematics\
    Rutgers University\
    New Brunswick, NJ 08854, USA
author:
- 'Andrzej Ros[ł]{}anowski'
- Saharon Shelah
title: 'Historic forcing for ${{\rm Depth}}$'
---

Introduction
============

The present paper is concerned with forcing a Boolean algebra which has some prescribed properties of ${{\rm Depth}}$. Let us recall that, for a Boolean algebra ${{\mathbb B}}$, its depth is defined as follows: $$\begin{array}{lcl}
{{\rm Depth}}({{\mathbb B}})&=&\sup\{|X|: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the Boolean
ordering}\;\},\\ 
{{\rm Depth}}^+({{\mathbb B}})&=&\sup\{|X|^+: X\subseteq{{\mathbb B}}\mbox{ is well-ordered by the
Boolean ordering}\;\}. 
  \end{array}$$ (${{\rm Depth}}^+({{\mathbb B}})$ is used to deal with attainment properties in the definition of ${{\rm Depth}}({{\mathbb B}})$, see e.g. [@RoSh:534 §1].) The depth (of Boolean algebras) is among cardinal functions that have more algebraic origins, and their relations to 