CompBoolAlg

**$Comp Bool Alg$** is the category whose objects are complete boolean algebras and whose morphisms are complete lattice homomorphisms, that is functions which preserve all joins and meets (including the bottom and top elements); it follows that they preserve the boolean negation operation. $Comp Bool Alg$ is a subcategory of Pos.

$Comp Bool Alg$ is given by a variety of algebras, or equivalently by an algebraic theory, so it is an equationally presented category; however, it requires operations of arbitrarily large arity. In fact, it is *not* a monadic category (over Set), because it lacks some free objects. Specifically, the **free complete boolean algebra** on a set $X$, while it exists (by general abstract nonsense) as a class, is small only if $X$ is finite (in which case it is finite and equals the free boolean algebra $2^{2^X}$ on $X$).

For all practical purposes, $Comp Bool Alg$ is not available in predicative mathematics. The definitions go through, but we cannot prove that it has any infinite objects. (More precisely, the power class of a nontrivial small complete boolean algebra must also be small.) Generally speaking, predicative mathematics treats infinite complete boolean algebras only as large objects.

In impredicative constructive mathematics, it no longer holds that the free complete boolean algebra on a finite set equals the free boolean algebra on that set. In particular, the free complete boolean algebra on the empty set is the set of truth values, while the free boolean algebra on the empty set is the set $\{\bot, \top\}$ of decidable truth values. (I'm not sure whether the free complete boolean algebra a positive finite number of elements is even small.)

In predicative constructive mathematics, even the free complete boolean algebra on the empty set is a proper class.

category: category

Revised on August 3, 2010 03:37:08
by Toby Bartels
(75.88.74.149)