Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra*
نویسنده
چکیده
Let A := (A,→, 1) be a Hilbert algebra. The monoid of all unary operations on A generated by operations αp : x → (p → x), which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of A. This semilattice is isomorphic to the semilattice of finitely generated filters of A, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of A. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of A, and the embedding of A into this extension preserves all existing joins and certain “compatible” meets.
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