Geodesic
Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 41-51.

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Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, 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.
Classification : 03G25, 06A12, 06A15, 08A35
Keywords: adjoint semilattice; Brouwerian extension; closure endomorphism; compatible meet; filter; Hilbert algebra; implicative semilattice; subtraction 
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Cīrulis, Jānis. Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 2, pp. 41-51. http://geodesic.mathdoc.fr/item/AUPO_2012__51_2_a3/