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Priestley style duality for distributive meet-semilattices. (English) Zbl 1238.03050

The authors generalize Priestley duality for distributive lattices to a duality for distributive meet-semilattices. A structure \(X= \langle X, \tau, \leq, X_0\rangle\) is called a generalized Priestley space if
1. \(\langle X, \tau, \leq \rangle\) is a Priestley space.
2. \(X_0\) is a dense subset of \(X\).
3. For each \(x\in X\) there is \(y\in X_0\) such that \(x\leq y\).
4. \(x\in X_0\) if and only if \(\mathcal{I}_x\) is up-directed.
5. For all \(x,y\in X\), \(x\leq y\) iff \((\forall U \in X^*)(x\in U \Rightarrow y\in U)\), where \(X^*\) is the set of \(X_0\)-admissible clopen up-sets of \(X\),
where \(\mathcal{I}_x = \{ U \,|\, x\not\in U \;\text{and} \;U \;\text{is \;an} \;X_0 \text{-admissible \;clopen \;upset \;of } \;X\}\).
Concerning the bounded distributive meet-semilattice, they prove:
Theorem 5.8 (Representation Theorem). For each bounded distributive meet-semilattice \(L\), there exists a generalized Priestley space \(X\) such that \(L \cong X^*\).
For generalized Priestley spaces \(X\) and \(Y\), a relation \(R\subseteq X\times Y\) is called a generalized Priestley morphism if the following two conditions are satisfied:
1. If not \(xRy\), then there is \(U\in Y^*\) such that \(y\not\in U\) and \(R[x]\subseteq U\).
2. If \(U\in Y^*\), then \(\square_R U \in X^*\), where \(\square_R U = \{ x\in X\,|\, (\forall y\in Y) (xRy \Rightarrow y\in U)\} = \{ x\in X \,|\, R[x]\subseteq U \}\).
Then, for the category GPS of generalized Priestley spaces and generalized Priestley morphisms, they prove:
Theorem 6.9. The category BDM is dually equivalent to the category GPS, where BDM is the category of bounded distributive meet-semilattices and meet-semilattice homomorphisms preserving top.
The result above is extend to non-bounded cases. A structure \(X= \langle X, \tau, \leq, X_0 \rangle\) is called a \(*\)-generalized Priestley space if
1. \(\langle X, \tau, \leq \rangle\) is a Priestley space.
2. \(X_0\) is a dense subset of \(X\).
3. \(x\in X_0\) iff \(\mathcal{I}_x\) is non-empty and up-directed.
4. For all \(x,y\in X\), \(x\leq y\) iff \((\forall U \in X^*)(x\in U \Rightarrow y\in U)\).
For the category GPS\(^*\) of \(*\)-generalized Priestley spaces and generalized Priestley morphisms, they prove the following result:
Theorem 9.2. The category DM is dually equivalent to the category GPS\(^*\), where DM is the category of distributive meet-semilattices and meet-semilattice homomorphisms.

MSC:

03G25 Other algebras related to logic
06A12 Semilattices
06D50 Lattices and duality
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