Vol. 15, No. 4, 1965

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
Free complete extensions of Boolean algebras

George Wesley Day

Vol. 15 (1965), No. 4, 1145–1151
Abstract

From considering questions about the existence of free α-complete Boolean algebras and free complete Boolean algebras, one is led naturally to the following problem: Given a Boolean algebra B, is it possible to embed B as a subalgebra in a complete Boolean algebra B in such a way that homomorphisms of B into complete Boolean algebras can be extended to complete homomorphisms on B? In general, the answer is “no”; this paper establishes that B can be so embedded if and only if every homomorphic image of B is atomic. Severaf other equivalent conditions on B are also developed.

To express these ideas more precisely, we say that the complete Boolean algebra B is a free complete extension of the Boolean B provided that there exist an isomorphism i of B into B such that

(i) if h is a homomorphism of B into a complete Boo# ean algebra C, then there is a complete homomorphism h of B into C such that hi = h;

(ii) B has no regular complete proper subalgebra which contains i[B]—that is, i[B] completely generates B. A Boolean algebra B is said to be superatomic if every homomorphic image of B is atomic (or, equivalently, if every subalgebra of B is atomic). Our principal result, then, is that a Boolean algebra B has a free complete extension if and only if B is superatomic.

Mathematical Subject Classification
Primary: 06.60
Milestones
Received: 2 June 1964
Published: 1 December 1965
Authors
George Wesley Day