In this paper we consider a general way of constructing profinite struc- tures 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.
The following situation is very popular in computer science: the expressive power of a countable set of syntactical objects Φ is studied over a countable family of structures W .
The following examples illustrate this situation:
- W = A * and Φ = {A * → M : M is a finite monoid}, 2. W are finite trees and Φ = {ϕ : ϕ is a first order sentence}, 3. W is a set of all finite trees and Φ is a family of all deterministic bottom-up tree automata, 4. W are all finite graphs and Φ is the family of all FO or MSO formulas.
One of the natural ideas how to represent recognition is to treat a syntactical object ϕ ∈ Φ as a function ϕ : W → K ϕ to a finite set K ϕ . Such a function recognises sets of objects of the form ϕ -1 (V ) ⊆ W , for V ⊆ K ϕ . 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 W and of recognisers Φ. Using adequate topology on W we show how to define a profinite structure W extending W . Moreover we prove that its possible to extend recognisers ϕ ∈ Φ to all profinite objects w ∈ W .
The following theorem is the main result.
Theorem 1.1. A family M 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 W = Σ * and Φ 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 Stone-Priestley 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 W . Using this topology we extend the space to a profinite structure and study the particular algebra of recognisable sets.
Additionally two properties must hold: a) Each object w ∈ W is totally described by some recogniser. That is for every object w ∈ W there is some recogniser ϕ ∈ Φ such that ϕ(w) = ϕ(w ′ ) for w ′ = w.
It is easy to check that all the examples from the introduction satisfy both axioms so they form frameworks.
Fix a framework (W , Φ).
Definition 2.2. A language L ⊆ W is called regular or recognisable if there exists ϕ ∈ Φ and V ⊆ K ϕ such that
The family of all regular languages is denoted as reg(W ). List all recognisers in Φ in a sequence Φ = {ϕ 0 , ϕ 1 , . . .}. It is good to think that recognisers appearing further in the sequence are more complicated. Let K i = K ϕ i and let X = i∈N K i . Because K i ’s are finite, X is a homeomorphic copy of Cantors discontinuum. Let µ : W → X be defined as follows µ(w) = (ϕ 0 (w), ϕ 1 (w), ϕ 2 (w), . . .) .
In other words µ maps an object w ∈ W to a sequence of values of all recognisers on that object. It is easy to see that µ is 1-1 because of Property a).
Definition 2.4. Let W = µ(W ) ⊆ X. The elements of the set W are called profinite objects of the framework (W , Φ). The topology on W is defined as a topology induced from X.
Since the order of coordinates in Cantors discontinuum does not affect its topology, we obtain the following fact.
Fact 2.5. The construction described above does not depend on the order of recognisers in the sequence ϕ 0 , ϕ 1 , . . ..
Proposition 2.6.
The natural embedding is µ.
W is a closed subset of a compact space, so it is compact.
This is an easy consequence of property a) of the framework.
Note that recognisers naturally extend to W . Proposition 2.7. For each recogniser ϕ i ∈ Φ we can extend it to all profinite objects w ∈ W ⊆ K i by an equation
For w ∈ W this definition is consistent with the original one.
The following lemma enables us to define regular languages from the topological point of view.
Moreover L → L is an isomorphism of the Boolean algebra of regular languages in W and the Boolean algebra of clopen subsets of W .
Proof. 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.
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 X to be a compact space and let B denote the Boolean algebra of clopen subsets of X. Definition 3.1. A sublattice of B is any subset M ⊆ B closed under union and inte
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