Abstract
In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.
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Presented by M. Ploscica.
The research of the first author was supported by the CONICET under grant no. 112-200801-02543.
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Celani, S.A., Montangie, D. Hilbert algebras with supremum. Algebra Univers. 67, 237–255 (2012). https://doi.org/10.1007/s00012-012-0178-z
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DOI: https://doi.org/10.1007/s00012-012-0178-z