Publication details

Residuated Operators and Dedekind–MacNeille Completion

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Year of publication 2021
Type Chapter of a book
MU Faculty or unit

Faculty of Science

Description 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 P is completed to its Dedekind–MacNeille completion DM(P) then the complete lattice DM(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 P and if DM(P) is an orthomodular lattice then the transformed lattice terms circled dot and -> form a left residuation in DM(P). However, it is a problem to determine when DM(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 P, the Dedekind–MacNeille completion DM(P) is an orthomodular lattice if and only if P is strongly D-continuous.
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