Modal Hilbert Algebras and their Triple Representation
Jānis Cīrulis
A modal Hilbert algebra is a Hilbert algebra equipped with a so called modal operator—a unary operation π which can be characterized by the identity π𝑥 → π𝑦 = 𝑥 → π𝑦. Basic properties of modal operators and algebras are presented; specified is a subclass of modal algebras called admissible. A Hilbert triple (𝐶, 𝐷, 𝑓 ) consists of Hilbert algebras 𝐶, 𝐷 and a certain mapping 𝑓 from 𝐶 to the endomorphism monoid of 𝐷. Every modal algebra determines a triple, and every triple gives rise to an admissible modal algebra (the triple construction). The arising correspondence between triples and admissible modal algebras is one-to-one up to isomorphisms; in particular, every admissible modal algebra is isomorphic to the algebra arising in this way from its triple (the strong triple representation theorem). Moreover, the correspondence is extended to a natural equivalence of the respective categories.
Keywords: Closure endomorphism, compatible meet, Glivenko operator, Hilbert algebra, modal algebra, modal operator, quasi-decomposition, triple, triple representation