Graph spaces and $\bot$-free Boolean algebras

Graph Spaces and 1-free Boolean Algebras

Let X denote an arbitrary second-countable, compact, zero-dimensional space. Our main result says that A' is a graph space, i.e., homeomorphic to the space of all complete subgraphs of a suitable graph. We first characterize graph spaces in terms of the Boolean algebras of their clopen subsets. Then it is proved that each countable Boolean algebra has the corresponding property. As a corollary ...

Representing free Boolean algebras

Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with א1 generators is represented. It w...

Free skew Boolean algebras

Boolean algebras, Stone spaces and TGD

The Facebook discussion with Stephen King about Stone spaces led to a highly interesting development of ideas concerning Boolean, algebras, Stone spaces, and p-adic physics. I have discussed these ideas already earlier but the improved understanding of the notion of Stone space helped to make the ideas more concrete. The basic ideas are briefly summarized. p-adic integers/numbers correspond to ...

Extensions of Free Boolean Algebras

In a series of papers [2, 3, 4, 5], B. Efimov has given sufficient conditions for a Boolean algebra to have a free subalgebra of given cardinality. For example, he has shown that if an infinite Boolean algebra A has cardinality greater than exp exp exp v, but the supremum of the cardinalities of subsets of disjoint elements in A is not greater than v, then A has a free subalgebra with (exp v) i...