Notes on Hilbert Lattices
Aldo Figallo-Orellano · 2011
Hilbert algebras started to be studied in the 50's and they consti- tute the algebraic counterpart of the implicative fragment of the propo- sitional implicative calculus ((13)). Later, Figallo, Ramand Saad ((9)), studied distributive Hilbert algebras (or dH-algebras), these alge- bras are Hilbert algebras which are bounded distributive lattices with respect to their natural order. In this work, we introduced the notion of Hilbert lattices (H-lattice) as dH-algebras dropping the distributy condition and the existence of the zero element 0. We considered pure Hilbert lattices (pH-lattice) which are a special class of H-lattices. These algebras are a partilurar case of order algebras studied by Chajda ans Halaus (5). We introduced the concept of ideal for pH-lattice; and we proved that every congruence of an arbitrary pH-lattice is a Rees congruence and that every pH-lattice is a Rees ideal algebra. Besides, we characterized some subdirectly irreducible and simple pH-lattices.