| Title:
|
A binary operation-based representation of a lattice (English) |
| Author:
|
Yettou, Mourad |
| Author:
|
Amroune, Abdelaziz |
| Author:
|
Zedam, Lemnaouar |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
252-272 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new constructed lattices. (English) |
| Keyword:
|
lattice |
| Keyword:
|
binary operation |
| Keyword:
|
neutral element |
| Keyword:
|
lattice representation |
| MSC:
|
06B05 |
| MSC:
|
06B15 |
| idZBL:
|
Zbl 07144937 |
| idMR:
|
MR4014586 |
| DOI:
|
10.14736/kyb-2019-2-0252 |
| . |
| Date available:
|
2019-09-30T14:59:39Z |
| Last updated:
|
2020-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147836 |
| . |
| Reference:
|
[1] Ashraf, M., Ali, S., Haetinger, C.: On derivations in rings and their applications..Aligarh Bull. Math. 25 (2006), 79-107. MR 2537802 |
| Reference:
|
[2] Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic..Springer, Berlin 2013. Zbl 1271.03001, MR 3024762 |
| Reference:
|
[3] Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners..Springer, Heidelberg 2007. |
| Reference:
|
[4] Birkhoff, G.: Lattice Theory. Third edition..Amer. Math. Soc., Providence 1967. MR 0227053 |
| Reference:
|
[5] Blyth, T. S.: Set theory and abstract algebra..Longman, London, New York 1975. MR 0223196 |
| Reference:
|
[6] Cooman, G. D., Kerre, E. E.: Order norms on bounded partially ordered sets..J. Fuzzy Math. 2 (1994), 281-310. Zbl 0814.04005, MR 1280148 |
| Reference:
|
[7] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Second edition..Cambridge University Press, 2002. MR 1902334, 10.1017/cbo9780511809088 |
| Reference:
|
[8] Dummit, D. S., Foote, R. M.: Abstract Algebra. Third edition..Hoboken, Wiley 2004. MR 2286236 |
| Reference:
|
[9] Ferrari, L.: On derivations of lattices..Pure Math. Appl. 12 (2001), 365-382. MR 1943869 |
| Reference:
|
[10] Grätzer, G., Wehrung, F.: Lattice Theory: Special Topics and Applications. Volume 1..Springer International Publishing Switzerland, 2014. MR 2451139, 10.1007/978-3-319-06413-0 |
| Reference:
|
[11] Grätzer, G., Wehrung, F.: Lattice theory: Special Topics and Applications. Volume 2..Springer International Publishing Switzerland, 2016. MR 2451139, 10.1007/978-3-319-44236-5 |
| Reference:
|
[12] Halaš, R., Pócs, J.: On the clone of aggregation functions on bounded lattices..Inform. Sci. 329 (2016), 381-389. 10.1016/j.ins.2015.09.038 |
| Reference:
|
[13] Jwaid, T., Baets, B. De, Kalická, J., Mesiar, R.: Conic aggregation functions..Fuzzy Sets Systems 167 (2011), 3-20. MR 2765243, 10.1016/j.fss.2010.07.004 |
| Reference:
|
[14] Karaçal, F., glu, M. N. Kesicio\v: A t-partial order obtained from t-norms..Kybernetika 47 (2011), 300-314. MR 2828579 |
| Reference:
|
[15] Karaçal, F., Mesiar, R.: Aggregation functions on bounded lattices..Int. J. General Systems 46 (2017), 37-51. MR 3623328, 10.1080/03081079.2017.1291634 |
| Reference:
|
[16] Kolman, B., Busby, R. C., Ross, S. C.: Discrete Mathematical Structures. Fourth edition..Prentice-Hall, Inc., 2003. |
| Reference:
|
[17] Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and theirs classification..Fuzzy Sets Systems 175 (2011), 48-56. MR 2803411, 10.1016/j.fss.2011.01.015 |
| Reference:
|
[18] Lidl, R., Pilz, G.: Applied Abstract Algebra. Second edition..Springer-Verlag, New York, Berlin, Heidelberg 1998. MR 1485777, 10.1007/978-1-4757-2941-2 |
| Reference:
|
[19] Lipschutz, S.: Discrete Mathematics. Third edition..McGraw-Hill, 2007. 10.1201/b13782 |
| Reference:
|
[20] Martínez, R., Massó, J., Neme, A., Oviedo, J.: On the lattice structure of the set of stable matchings for a many to one model..Optimization 50 (2001), 439-457. MR 1892915, 10.1080/02331930108844574 |
| Reference:
|
[21] Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices..Fuzzy Sets and Systems 202 (2012), 75-88. MR 2934787, 10.1016/j.fss.2012.03.002 |
| Reference:
|
[22] Mesiar, R., Komorníková, M.: Aggregation functions on bounded posets..35 Years of Fuzzy Set Theory, Springer, Berlin, Heidelberg 261 (2010), pp. 3-17. MR 3289892, 10.1007/978-3-642-16629-7_1 |
| Reference:
|
[23] Ponasse, D., Carrega, J. C.: Algèbre et tobologie boléennes..Masson, Paris 1979. MR 0532013 |
| Reference:
|
[24] Risma, E. P.: Binary operations and lattice structure for a model of matching with contracts..Math. Soc. Sci. 73 (2015), 6-12. MR 3294778, 10.1016/j.mathsocsci.2014.11.001 |
| Reference:
|
[25] Roman, S.: Lattices and Ordered Sets..Springer Science and Business Media, New York 2008. MR 2446182, 10.1007/978-0-387-78901-9 |
| Reference:
|
[26] Rosenfeld, A.: An Introduction to Algebraic Structures..Holden-Day, San Francisco 1968. MR 0232630 |
| Reference:
|
[27] Schröder, B. S.: Ordered Sets..Birkhauser, Boston 2003. MR 1944415, 10.1007/978-1-4612-0053-6 |
| Reference:
|
[28] Szász, G.: Translationen der verbände..Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961), 449-453. MR 0132705, 10.1007/bf01238704 |
| Reference:
|
[29] Szász, G.: Derivations of lattices..Acta Sci. Math. 37 (1975), 149-154. MR 0382090, 10.1016/s0020-1693(00)93924-0 |
| Reference:
|
[30] Xin, X. L., Li, T. Y., Lu, J. H.: On derivations of lattices..Inform. Sci. 178 (2008), 307-316. MR 2363221, 10.1016/j.ins.2007.08.018 |
| . |
