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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
