DISCRETE DUALITY FROM THE CLASSICAL CASE TO THE NON-DISTRIBUTIVE CASE

Boolean Algebra

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Boolean Algebra Distributive Lattices Descriptive General F rames with Operators

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drop negation Relational P riestley Boolean Space & add operators Spaces

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Distributive Lattices

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We have already exploited some of these studies in the previous chapter, particularly the duality for dis- tributive lattices expanded with modal operations as presented in [Conradie & Palmigiano 2012]. But a duality theory has also started to be developed for bounded lattices which are no longer distributive (for instance,[Hartung 1992, Urquhart 1978] and for the modal view on it [Haim 2000]), and nally for partially ordered sets along with a development of canonical extension theory for order-preserving and order reversing expansions [Dunn, Gehrke & Palmigiano 2005, Gehrke 2006]. We rst provide (4.2) a brief overview of the discrete duality that arise on increasingly generalized modal settings, to nally focus on the canonical extension and discrete duality for the non-distributive setting (4.3.), as preparation for the completeness result exposition in Chapter 5.

4.2. Discrete duality from the classical case to the non-distributive case.

A brief overview on how the discrete duality emerges in the classical and distributive settings will show that the non-distributive discrete duality comes as a natural extension of a well-known strategy.

4.2.1. Stone duality. The Stone representation theorem states that every Boolean algebra Acan be

embedded in the complete and atomic Boolean algebraAσ dened as the power set algebra of the collection

of its ultralters1_{, where the latter generalize the notion of atoms. Thus even when a Boolean algebra is not} atomic nor complete, it can be embedded into a complete and atomic Boolean algebra, as every power-set algebra belongs to this category. In fact, this is often reformulated as an isomorphism (every Boolean algebra is a power-set algebra, up to isomorphism). Thus, a very obvious advantage of Representation theorems -of which this one is an example- is that they allow to reduce a class of structures to a proper subclass with a simpler behaviour. The usual shape of this type of theorems is that every element of the class of structuresS

is isomorphic to some element of a proper subclassS0 ⊂S of structures. It is often the case that this subclass S0 has some nice extra properties that makes it more suitable to work with than the original classS, while

the isomorphism ensures that any isomorphim-invariant result proved in S0 will carry over toS. But this

is just one aspect of the insight gained, a fully detailed formulation of a representation theorem provides a precise denition of the embedding used to prove it.

4.2. DISCRETE DUALITY FROM THE CLASSICAL CASE TO THE NON-DISTRIBUTIVE CASE. 54 Theorem 73. Stone representation theorem. LetA=hA,∧,∨,¬,0,1ibe a Boolean algebra and letU f(A)be

the set of ultralters ofA. Then the functionv:A−→℘(U f(A))dened byv(a) = ˆa={U ∈U f(A)|a∈U}

is an injective morphism of lattices. Consequently, every Boolean algebraAcan be embedded in the complete

and atomic Boolean algebra Aσ dened as the powerset algebra of the set of ultralters of A

Such embedding provides a concrete and intuitive interpretation of otherwise abstractly dened operations. More precisely, this result links the rather abstract meaning of Boolean operations in A to a concrete (i.e.

set theoretic) view of such meaning. Thusa∧bcan be seen asa[∧b= ˆa∩ˆb, for instance, with abstract meet

now being interpreted as set intersection.

The main interest of representation theorems and canonical extensions in the context of this thesis, and more generally from logical viewpoint, is that they are an important source of completeness results. For instance, via the reformulation in terms of isomorphism, Stone representation theorem presents the completeness of classical logic as an essentially algebraic result. We can directly see Aas a subalgebra of Aσ by recovering

the fact that, given the embedding, A is isomorphic to its image underv(·). Since validity of equations is

preserved under taking subalgebras, we can easily see this representation theorem as a powerful tool to obtain completeness results via discrete duality (validity of equations is preserved on taking subalgebras, thus logical counterexamples in the Lindenbaum algebra are preserved in the opposite direction: they are still invalid in the canonical extension of the Lindenbaum algebra).

The existence of the following two dualities of categories is crucial for the Stone representation theorem.

CABA (·)₊ // Sets (·)+ oo Discrete duality BA (·)_∗ // (·)σ OO (·)_•=U◦(·)_∗ == Stone U OO (·)∗ oo T opological duality

Given a Boolean algebraA, its associated stone space(A)∗is formed by taking the subset of its powerset that

consists only of ultralters and setting as the basis of the space the collectionS

a∈A{U ∈U f(A)|a∈U} of

families of ultralters selected by a common element. In general, the collection of clopens of any topological space forms a Boolean algebra, but the particular Boolean algebra that arises from a given Stone space has a special property with respect to the original BA -and this is what the Stone representation theorem states-: every Boolean algebra A is isomorphic to the Boolean algebra ((A)∗)

∗ _{of clopen subsets of its associated}

4.2. DISCRETE DUALITY FROM THE CLASSICAL CASE TO THE NON-DISTRIBUTIVE CASE. 55