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Published in:
30-01-2020 | Focus
On residuation in paraorthomodular lattices
Published in: Soft Computing | Issue 14/2020
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Abstract
Paraorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution \({\mathbf {A}}\) can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever \({\mathbf {A}}\) is distributive.