Residuated Operators and Dedekind–MacNeille Completion

Abstract

The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset \({\mathbf{P}}\) is completed to its Dedekind–MacNeille completion \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\) then the complete lattice \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\) becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. A more complicated situation is with orthomodular and pseudo-orthomodular posets. We show which operators M (multiplication) and R (residuation) yield operator left-residuation in a pseudo-orthomodular poset \({\mathbf{P}}\) and if \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\) is an orthomodular lattice then the transformed lattice terms \(\odot \) and \(\rightarrow \) form a left residuation in \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\). However, it is a problem to determine when \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\) is an orthomodular lattice. We get some classes of pseudo-orthomodular posets for which their Dedekind–MacNeille completion is an orthomodular lattice and we introduce the so-called strongly D-continuous pseudo-orthomodular posets. Finally we prove that, for a pseudo-orthomodular poset \({\mathbf{P}}\), the Dedekind–MacNeille completion \({{\,\mathrm{\mathbf {DM}}\,}}(\mathbf{P})\) is an orthomodular lattice if and only if \({\mathbf{P}}\) is strongly D-continuous.