Residuated Operators and Dedekind–MacNeille Completion
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F21%3A00118809" target="_blank" >RIV/00216224:14310/21:00118809 - isvavai.cz</a>
Alternative codes found
RIV/61989592:15310/21:73609450
Result on the web
<a href="https://doi.org/10.1007/978-3-030-52163-9_5" target="_blank" >https://doi.org/10.1007/978-3-030-52163-9_5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/978-3-030-52163-9_5" target="_blank" >10.1007/978-3-030-52163-9_5</a>
Alternative languages
Result language
angličtina
Original language name
Residuated Operators and Dedekind–MacNeille Completion
Original language 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.
Czech name
—
Czech description
—
Classification
Type
C - Chapter in a specialist book
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Book/collection name
Algebraic Perspectives on Substructural Logics
ISBN
9783030521622
Number of pages of the result
16
Pages from-to
57-72
Number of pages of the book
193
Publisher name
Springer
Place of publication
Cham
UT code for WoS chapter
—