Completions of BOOLEAN Algebras with operators

✍ Scribed by J. Donald Monk

• Publisher: John Wiley and Sons
• Year: 1970
• Tongue: English
• Weight: 401 KB
• Volume: 46
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19700460105

✦ Synopsis

The notion of a BooLEan algebra with operators was introduced by J~NSSON and TARSKI [ 5 ] . It encompasses as special cases relation algebras (TARSKI [9]), closure algebras (MCKINSEY-TARSKI [S]), cylindric algebras (HENRIN-TARSKI [4]), polyadic algebras (HALMOS [Z]), and other algebras which have been studied in recent years. One of the basic results of [5]

is that any Booman algebra with operators can be extended to one that is complete and atomic. The extension does not preserve any Booman sums (joins) which are essentially infinite, however. It is the main purpose of this paper to describe a completion that, while not atomic in general, does preserve all sums (and products).

In section 1 the theory of such completions is extensively developed, patterning the development after section 2 of [ 5 ] . It turns out that the proofs are much simpler than in [5], so they are given only briefly. The second short section of the paper deals briefly with completions of some of the special kinds of algebras mentioned in the preceding paragraph.

We adopt the notation of [5], with the following exceptions and additions.