## 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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### References:

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