Equational theories of profinite structures
Streszczenie
In this paper we consider a general way of constructing profinite structures based on a given framework — a countable family of objects and a countable family of recognisers (e.g. formulas). The main theorem states:
A subset of a family of recognisable sets is a lattice if and only if it is definable by a family of profinite equations.
This result extends Theorem 5.2 from [GGEP08] expressed only for finite words and morphisms to finite monoids.
One of the applications of our theorem is the situation where objects are finite relational structures and recognisers are first order sentences. In that setting a simple characterisation of lattices of first order formulas arise.
1 Introduction
The following situation is very popular in computer science: the expressive power of a countable set of syntactical objects is studied over a countable family of structures .
The following examples illustrate this situation:

and ,

are finite trees and ,

is a set of all finite trees and is a family of all deterministic bottomup tree automata,

are all finite graphs and is the family of all FO or MSO formulas.
One of the natural ideas how to represent recognition is to treat a syntactical object as a function to a finite set . Such a function recognises sets of objects of the form
for . Such sets are usually called regular or recognisable.
All the examples presented above fall into this schema: formula is a function to , homomorfizm is a function to a finite monoid, deterministic automaton maps a tree into the state reached at the root.
In this paper we work with a very general setting of families of objects and of recognisers . Using adequate topology on we show how to define a profinite structure extending . Moreover we prove that its possible to extend recognisers to all profinite objects .
The following theorem is the main result.
A family of recognisable sets of objects is a lattice if and only if it is defined by a set of profinite equations.
In paper [GGEP08] authors show analogous theorem in the context of profinite monoids. It is a special case of our result where and are homomorphisms to finite monoids.
Paper [GGEP10] is devoted to the idea of recognisers — particular functions defined on a given topological (or more generally uniform) space. Authors show that each Boolean algebra of subsets of a space has a minimal recogniser. Additionally various additional structures of the space (e.g. the structure of a monoid) provides additional properties of a recogniser. The key tool is the StonePriestley duality.
The approach presented in this paper is different. We start with a fixed family of recognisers and using them we provide adequate topology on . Using this topology we extend the space to a profinite structure and study the particular algebra of recognisable sets.
2 Profinite structures
{definition}A framework consists of

a countable set of objects,

a countable set of recognisers, i.e. functions where is a finite set.
Additionally two properties must hold:
 a)

Each object is totally described by some recogniser. That is for every object there is some recogniser such that for .
 b)

Recognisers are closed under intersections. That is for every recognisers and every sets of values there exists a recogniser and a set such that
It is easy to check that all the examples from the introduction satisfy both axioms so they form frameworks.
Fix a framework .
A language is called regular or recognisable if there exists and such that
The family of all regular languages is denoted as .
is a Boolean algebra.
Dowód.
By Property b) regular languages are closed under intersection. Of course they are also closed under complementation. So they form a Boolean algebra. ∎
List all recognisers in in a sequence . It is good to think that recognisers appearing further in the sequence are more complicated. Let and let . Because ’s are finite, is a homeomorphic copy of Cantors discontinuum. Let be defined as follows
In other words maps an object to a sequence of values of all recognisers on that object.
It is easy to see that is 11 because of Property a).
Let . The elements of the set are called profinite objects of the framework . The topology on is defined as a topology induced from .
Since the order of coordinates in Cantors discontinuum does not affect its topology, we obtain the following fact.
The construction described above does not depend on the order of recognisers in the sequence .
The following proposition summarises the properties of .

naturally embeds into ,

is a compact topological space,

for each the point is isolated,

is a countable dense subset of .
Dowód.

The natural embedding is .

is a closed subset of a compact space, so it is compact.

This is an easy consequence of property a) of the framework.

is a closure of , so is dense in .
∎
Note that recognisers naturally extend to .
For each recogniser we can extend it to all profinite objects by an equation
For this definition is consistent with the original one.
The following lemma enables us to define regular languages from the topological point of view.
A language is regular if and only if is an clopen subset of .
Moreover is an isomorphism of the Boolean algebra of regular languages in and the Boolean algebra of clopen subsets of .
Dowód.
Regular languages form a clopen base of the topology and they are closed under finite Boolean operations. In a compact space each clopen subset is a Boolean combination of the clopen base sets. ∎
3 Topology
In this section we provide a general characterisation of sublattices and Boolean subalgebras of the algebra of clopen subsets of a compact space. This whole theory is based only on topological properties and holds for any compact space.
Fix to be a compact space and let denote the Boolean algebra of clopen subsets of .
A sublattice of is any subset closed under union and intersection and containing and .
An equation is a formula for . We say that a subset satisfies an equation if the following property holds
For a given set of equations , the family of subsets that satisfy all equations in will be called a family of subsets defined by .
For a given compact space a family is a lattice if and only if it is defined by some set of equations.
Dowód.
Of course for a given set of equations, a family of clopen subsets satisfying them is a lattice.
Take any lattice . Let be a set of all equations satisfied by every set in . Take any subset satisfying . We will show that .
Let
If , then is a covering by open sets of a compact , so there is a finite family such that . So . Assume by contradiction that there exists .
Consider
If , then complements of elements in cover so there is a finite family such that . But then and , so a contradiction to the fact that . So there exists .
Consider equation and any such that . Then so by the definition so satisfies . Therefore . But does not satisfy . A contradiction. ∎
A family is a Boolean subalgebra if and only if it is defined by a symmetric set of equations .
4 Main theorem
Using the topological result from the previous section we can prove the main theorem from the introduction.
A family of recognisable sets of objects is a lattice if and only if it is defined by a set of profinite equations.
Firstly we adopt the definition of equation to the case of regular languages. This definition follows the one from [GGEP08].
We say that a regular language satisfies an equation for if
In other words language recognised by a recogniser and a set of values satisfies iff
Proof of Theorem 1.
Take any subfamily of regular languages and consider
By Lemma 2 and is a lattice if and only if is a lattice. Additionally is defined by a set of equations in the meaning of Definition 4 if and only if is defined by a set of equations in the meaning of Definition 3.
But is defined by a set of equations of and only if it is a lattice, because of Theorem 3 and the fact that is a compact topological space. ∎
A family of recognisable languages of objects is a Boolean algebra if and only if it is defined by a set of profinite symmetric equations.
5 Conclusions
In this section we propose various applications of the main theorem.
Firstly consider a case when and consists of all homomorphisms of to finite monoids. In that case is just a space of profinite words and Theorem 1 coincides with Theorem 5.2 from [GGEP08].
It turns out that Theorem 1 gives some insight into the structure of lattices of first order formulas. Fix a relational signature and consider as a family of all finite structures over . Let be a set of all FO sentences over . Of course is a framework. We denote this framework as first order framework over signature .
If is a first order framework then for every profinite object there exists a structure over such that for all FO sentences
Dowód.
An easy application of the compactness theorem. ∎
Of course there are infinite structures that don’t represent any element of . For example take and let express that is a linear order and that . Then is a model for but no finite structure satisfies it. So does not represent any element of .
Every lattice of first order formulas is defined by some family of implications , where are (potentially infinite) structures.
We say that a formula satisfies an implication iff
Dowód.
5.1 Acknowledgements
The author would like to thank Mikołaj Bojańczyk, Damian Niwiński and Henryk Michalewski for their helpful comments.
Literatura
 [GGEP08] Mai Gehrke, Serge Grigorieff, and Jean Éric Pin. Duality and equational theory of regular languages. In ICALP, pages 246–257, 2008.
 [GGEP10] Mai Gehrke, Serge Grigorieff, and Jean Éric Pin. A topological approach to recognition. In Automata, Languages and Programming, volume 6199 of LNCS, pages 151–162. Springer Berlin / Heidelberg, 2010.