On hyperplanes and semispaces in max–min convex geometry

Viorel Nitica; Sergeĭ Sergeev

Kybernetika (2010)

  • Volume: 46, Issue: 3, page 548-557
  • ISSN: 0023-5954

Abstract

top
The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.

How to cite

top

Nitica, Viorel, and Sergeev, Sergeĭ. "On hyperplanes and semispaces in max–min convex geometry." Kybernetika 46.3 (2010): 548-557. <http://eudml.org/doc/196930>.

@article{Nitica2010,
abstract = {The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.},
author = {Nitica, Viorel, Sergeev, Sergeĭ},
journal = {Kybernetika},
keywords = {tropical convexity; fuzzy algebra; separation; tropical convexity; fuzzy algebra; separation},
language = {eng},
number = {3},
pages = {548-557},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On hyperplanes and semispaces in max–min convex geometry},
url = {http://eudml.org/doc/196930},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Nitica, Viorel
AU - Sergeev, Sergeĭ
TI - On hyperplanes and semispaces in max–min convex geometry
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 3
SP - 548
EP - 557
AB - The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.
LA - eng
KW - tropical convexity; fuzzy algebra; separation; tropical convexity; fuzzy algebra; separation
UR - http://eudml.org/doc/196930
ER -

References

top
  1. Birkhoff, G., Lattice Theory, American Mathematical Society, Providence, RI 1993. Zbl0537.06001
  2. Cechlárová, K., Eigenvectors in bottleneck algebra, Linear Algebra Appl. 175 (1992), 63–73. MR1179341
  3. Cohen, G., Gaubert, S., Quadrat, J. P., Singer, I., Max-plus convex sets and functions, In: Idempotent Mathematics and Mathematical Physics (G. Litvinov and V. Maslov, eds.), AMS, Providence 2005, pp. 105–129. E-print arXiv:math/0308166. Zbl1093.26005MR2149000
  4. Develin, M., Sturmfels, B., Tropical convexity, Documenta Math. 9 (2004), 1–27. E-print arXiv:math/0308254. Zbl1054.52004MR2054977
  5. Gaubert, S., Katz, R., 10.1007/11828563_13, In: Lecture Notes in Computer Science 4136, Springer, New York 2006. pp. 192–206. Zbl1134.52303MR2281601DOI10.1007/11828563_13
  6. Gaubert, S., Sergeev, S., 10.1007/s10958-008-9243-8, J. Math. Sci. 155 (2008), 6, 815–829. E-print arXiv:math/0706.3347. Zbl1173.47045MR2366235DOI10.1007/s10958-008-9243-8
  7. Gavalec, M., Periodicity in Extremal Algebras, Gaudeamus, Hradec Králové 2004. 
  8. Gavalec, M., Plávka, J., Strong regularity of matrices in general max-min algebra, Linear Algebra Appl. 371 (2003), 241–254. MR1997373
  9. Golan, J., Semirings and Their Applications, Kluwer, Dordrecht 2000. Zbl0947.16034MR1746739
  10. Litvinov, G. L., Maslov, V. P., Shpiz, G. B., Idempotent functional analysis: an algebraic approach, Math. Notes 69 (2001), 5, 758–797. Zbl1017.46034MR1846814
  11. Nitica, V., The structure of max-min hyperplanes, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.08.022. Zbl1180.52005MR2566489
  12. Nitica, V., Singer, I., 10.1080/02331930600819852, I. Optimization 56 (2007), 171–205. Zbl1127.52001MR2288512DOI10.1080/02331930600819852
  13. Nitica, V., Singer, I., 10.1080/02331930601123031, II. Optimization 56 (2007), 293–303. Zbl1127.52001MR2326254DOI10.1080/02331930601123031
  14. Nitica, V., Singer, I., 10.1016/j.laa.2007.09.032, I. Segments. Linear Algebra Appl. 428 (2008), 7, 1439–1459. Zbl1134.52300MR2388630DOI10.1016/j.laa.2007.09.032
  15. Nitica, V., Singer, I., Contributions to max-min convex geometry, II. Semispaces and convex sets. Linear Algebra Appl. 428 (2008), 8–9, 2085–2115. Zbl1134.52300MR2401643
  16. Sergeev, S. N., 10.1023/B:MATN.0000009021.18823.52, Math. Notes 74 (2003), 6, 848–852. Zbl1108.52301MR2054008DOI10.1023/B:MATN.0000009021.18823.52
  17. Zimmermann, K., A general separation theorem in extremal algebras, Ekonom.-Mat. Obzor 13 (1977), 179–201. Zbl0365.90127MR0453607
  18. Zimmermann, K., Convexity in semimodules, Ekonom.-Mat. Obzor 17 (1981), 199–213. Zbl0477.52002MR0629908

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.