Residuated Operators and Dedekind–MacNeille Completion

C 2021

Residuated Operators and Dedekind–MacNeille Completion

CHAJDA, Ivan, Helmut LÄNGER and Jan PASEKA

Basic information

Original name

Residuated Operators and Dedekind–MacNeille Completion

Authors

CHAJDA, Ivan (guarantor), Helmut LÄNGER and Jan PASEKA (203 Czech Republic, belonging to the institution)

Edition

Cham, Algebraic Perspectives on Substructural Logics, p. 57-72, 16 pp. TREN, volume 55, 2021

Publisher

Springer

Other information

Language

English

Type of outcome

Kapitola resp. kapitoly v odborné knize

Field of Study

10101 Pure mathematics

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

Publication form

printed version "print"

References:

URL

RIV identification code

RIV/00216224:14310/21:00118809

Organization unit

Faculty of Science

ISBN

978-3-030-52162-2

DOI

http://dx.doi.org/10.1007/978-3-030-52163-9_5

Keywords in English

residuated lattices; Operators

Abstract

V originále

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.

Links

GA18-06915S, research and development project

Name: Nové přístupy k agregačním operátorům v analýze a zpracování dat

Investor: Czech Science Foundation

Citovat

CHAJDA, Ivan, Helmut LÄNGER and Jan PASEKA. Residuated Operators and Dedekind–MacNeille Completion. In Fazio D., Ledda A., Paoli F. Algebraic Perspectives on Substructural Logics. Cham: Springer, 2021, p. 57-72. TREN, volume 55. ISBN 978-3-030-52162-2. Available from: https://dx.doi.org/10.1007/978-3-030-52163-9_5.

@inbook{1721136,
   author = {Chajda, Ivan and Länger, Helmut and Paseka, Jan},
   address = {Cham},
   booktitle = {Algebraic Perspectives on Substructural Logics},
   doi = {http://dx.doi.org/10.1007/978-3-030-52163-9_5},
   editor = {Fazio D., Ledda A., Paoli F.},
   keywords = {residuated lattices; Operators},
   howpublished = {tištěná verze "print"},
   language = {eng},
   location = {Cham},
   isbn = {978-3-030-52162-2},
   pages = {57-72},
   publisher = {Springer},
   title = {Residuated Operators and Dedekind–MacNeille Completion},
   url = {https://doi.org/10.1007/978-3-030-52163-9_5},
   year = {2021}
}

TY  - CHAP
ID  - 1721136
AU  - Chajda, Ivan - Länger, Helmut - Paseka, Jan
PY  - 2021
TI  - Residuated Operators and Dedekind–MacNeille Completion
VL  - TREN, volume 55
PB  - Springer
CY  - Cham
SN  - 9783030521622
KW  - residuated lattices
KW  - Operators
UR  - https://doi.org/10.1007/978-3-030-52163-9_5
L2  - https://doi.org/10.1007/978-3-030-52163-9_5
N2  - 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.
ER  -

CHAJDA, Ivan, Helmut LÄNGER and Jan PASEKA. Residuated Operators and Dedekind–MacNeille Completion. In Fazio D., Ledda A., Paoli F. \textit{Algebraic Perspectives on Substructural Logics}. Cham: Springer, 2021, p.~57-72. TREN, volume 55. ISBN~978-3-030-52162-2. Available from: https://dx.doi.org/10.1007/978-3-030-52163-9\_{}5.

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