## 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:

  Bezhanishvili, G., and R. Jansana, ’Duality for distributive and implicative semilattices’, Preprints of University of Barcelona Research Group in Non-Classical Logics. Available from http://www.mat.ub.edu/$$\sim$$logica/docs/BeJa08-m.pdf , 2008.  Bezhanishvili, G., and R. Jansana, ’Esakia style duality for implicative semilattices’, Applied Categorical Structures, to appear. · Zbl 1294.06005  Celani S.A.: ’Topological representation of distributive semilattices’. Sci. Math. Jpn. 58(1), 55–65 (2003) · Zbl 1041.06002  Erné M.: ’Choiceless, pointless, but not useless: dualities for preframes’. Appl. Categ. Structures 15(5-6), 541–572 (2007) · Zbl 1137.06001  Frink O.: ’Ideals in partially ordered sets’. Amer. Math. Monthly 61, 223–234 (1954) · Zbl 0055.25901  Gierz, G., K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott, Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge, 2003. · Zbl 1088.06001  Grätzer, G., General lattice theory, second ed., Birkhäuser Verlag, Basel, New appendices by the author with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung and R. Wille, 1998. · Zbl 0909.06002  Hansoul, G., ’Priestley duality for distributive semilattices’, Institut de Mathématique, Université de Liège, Preprint 97.011, 2003. · Zbl 1365.06004  Hansoul G., Poussart C.: ’Priestley duality for distributive semilattices’. Bull. Soc. Roy. Sci. Liège 77, 104–119 (2008) · Zbl 1365.06004  Horn A., Kimura N.: ’The category of semilattices’. Algebra Universalis 1(1), 26–38 (1971) · Zbl 0249.06004  Priestley H.A.: ’Representation of distributive lattices by means of ordered Stone spaces’. Bull. London Math. Soc. 2, 186–190 (1970) · Zbl 0201.01802  Priestley H.A.: ’Ordered topological spaces and the representation of distributive lattices’. Proc. London Math. Soc. (3) 24, 507–530 (1972) · Zbl 0323.06011  Stone M.: ’Topological representation of distributive lattices and Brouwerian logics’. Časopis pešt. mat. fys. 67, 1–25 (1937)
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