Show that every bounded chain is a stone algebra.
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Stone Algebras
Walter Guttmann
June 28, 2019
Abstract
A range of algebras between lattices and Boolean algebras generalise the notion of a complement. We develop a hierarchy of these
pseudo-complemented algebras that includes Stone algebras. Independently of this theory we study filters based on partial orders. Both
theories are combined to prove Chen and Gr¨atzer’s construction theorem for Stone algebras. The latter involves extensive reasoning about
algebraic structures in addition to reasoning in algebraic structures.
1 Synopsis and Motivation
This document describes the following four theory files:
∗ Lattice Basics is a small theory with basic definitions and facts extending Isabelle/HOL’s lattice theory. It is used by the following theories.
∗ Pseudocomplemented Algebras contains a hierarchy of algebraic structures between lattices and Boolean algebras. Many results of Boolean
algebras can be derived from weaker axioms and are useful for more
general models. In this theory we develop a number of algebraic structures with such weaker axioms. The theory has four parts. We first
extend lattices and distributive lattices with a pseudocomplement operation to obtain (distributive) p-algebras. An additional axiom of
the pseudocomplement operation yields Stone algebras. The third
part studies a relative pseudocomplement operation which results in
Heyting algebras and Brouwer algebras. We finally show that Boolean
algebras instantiate all of the above structures.
∗ Filters contains an order-/lattice-theoretic development of filters. We
prove the ultrafilter lemma in a weak setting, several results about the
lattice structure of filters and a few further results from the literature.
Our selection is due to the requirements of the following theory.
∗ Construction of Stone Algebras contains the representation of Stone
algebras as triples and the corresponding isomorphisms [7, 21]. It
is also a case study of reasoning about algebraic structures. Every
Stone algebra is isomorphic to a triple comprising a Boolean algebra,
a distributive lattice with a greatest element, and a bounded lattice
homomorphism from the Boolean algebra to filters of the distributive
2
lattice. We carry out the involved constructions and explicitly state
the functions defining the isomorphisms. A function lifting is used
to work around the need for dependent types. We also construct an
embedding of Stone algebras to inherit theorems using a technique of
universal algebra.
Algebras with pseudocomplements in general, and Stone algebras in particular, appear widely in mathematical literature; for example, see [4, 5, 6, 17].
We apply Stone algebras to verify Prim’s minimum spanning tree algorithm
in Isabelle/HOL in [20].
There are at least two Isabelle/HOL theories related to filters. The
theory HOL/Algebra/Ideal.thy defines ring-theoretic ideals in locales with
a carrier set. In the theory HOL/Filter.thy a filter is defined as a set of sets.
Filters based on orders and lattices abstract from the inner set structure;
this approach is used in many texts such as [4, 5, 6, 9, 17]. Moreover, it is
required for the construction theorem of Stone algebras, whence our theory
implements filters this way.
Besides proving the results involved in the construction of Stone algebras,
we study how to reason about algebraic structures defined as Isabelle/HOL
classes without carrier sets. The Isabelle/HOL theories HOL/Algebra/*.thy
use locales with a carrier set, which facilitates reasoning about algebraic
structures but requires assumptions involving the carrier set in many places.
Extensive libraries of algebraic structures based on classes without carrier
sets have been developed and continue to be developed [1, 2, 3, 10, 11, 13,
14, 15, 16, 19, 22, 24, 25, 26]. It is unlikely that these libraries will be
converted to carrier-based theories and that carrier-free and carrier-based
implementations will be consistently maintained and evolved; certainly this
has not happened so far and initial experiments suggest potential drawbacks
for proof automation [12]. An improvement of the situation seems to require
some form of automation or system support that makes the difference irrelevant.
In the present development, we use classes without carrier sets to reason
about algebraic structures. To instantiate results derived in such classes,
the algebras must be represented as Isabelle/HOL types. This is possible to
a certain extent, but causes a problem if the definition of the underlying set
depends on parameters introduced in a locale; this would require dependent
types.
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