Glasnik Matematicki, Vol. 55, No. 2 (2020), 177-190.

ON SOME PARTIAL ORDERS ON A CERTAIN SUBSET OF THE POWER SET OF RINGS

Gregor Dolinar, Bojan Kuzma, Janko Marovt and Burcu Ungor

University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25, SI-1000 Ljubljana, Slovenia
and
IMFM, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia
e-mail: gregor.dolinar@fe.uni-lj.si

University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia
and
IMFM, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia
e-mail: bojan.kuzma@upr.si

University of Maribor, Faculty of Economics and Business, Razlagova 14, SI-2000 Maribor, Slovenia
and
IMFM, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia
e-mail: janko.marovt@um.si

Ankara University, Faculty of Sciences, Department of Mathematics, 06100 Tandogan, Ankara, Turkey
e-mail: bungor@science.ankara.edu.tr

Abstract.   Let 𝓡 be a ring with identity and let 𝓙𝓡 be a collection of subsets of 𝓡 such that their left and right annihilators are generated by the same idempotent. % from 𝓡. We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to 𝓙𝓡, present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to 𝓙𝓡, show that they are indeed partial orders when 𝓡 is a Baer *-ring, and connect them with one-sided sharp and star partial orders.

2010 Mathematics Subject Classification.   06F25, 06A06, 15A09

Key words and phrases.   Baer *-ring, sharp partial order, core partial order, star partial order, one-sided partial order

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.2.01

References:

  1. J. Antezana, C. Cano, I. Mosconi and D. Stojanoff, A note on the star order in Hilbert spaces, Linear Multilinear Algebra 58 (2010), 1037-1051.
    MathSciNet     CrossRef

  2. J. K. Baksalary and S. K. Mitra, Left-star and right-star partial orderings, Linear Algebra Appl. 149 (1991), 73-89.
    MathSciNet     CrossRef

  3. O. M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (2010), 681-697.
    MathSciNet     CrossRef

  4. S. K. Berberian, Baer *-rings, Springer-Verlag, New York, 1972.
    MathSciNet    

  5. D. S. Djordjević, D. S. Rakić and J. Marovt, Minus partial order in Rickart rings, Publ. Math. Debrecen 87 (2015), 291-305.
    MathSciNet     CrossRef

  6. G. Dolinar, A. E. Guterman and J. Marovt, Monotone transformations on B(𝓗) with respect to the left-star and the right-star partial order, Math. Inequal. Appl. 17 (2014), 573-589.
    MathSciNet     CrossRef

  7. G. Dolinar and J. Marovt, Star partial order on $B(𝓗)$, Linear Algebra Appl. 434 (2011), 319-326.
    MathSciNet     CrossRef

  8. M. P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141.
    MathSciNet     CrossRef

  9. R. E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980), 1-13.
    MathSciNet    

  10. R. E. Hartwig and J. Luh, A note on the group structure on unit regular ring elements, Pacific J. Math. 71 (1977), 449-461.
    MathSciNet     CrossRef

  11. A. Herrero and N. Thome, Sharp partial order and linear autonomous systems, Appl. Math. Comput. 366 (2020), 124736, 11 pp.
    MathSciNet     CrossRef

  12. I. Kaplansky, Rings of operators, Benjamin, New York, 1968.
    MathSciNet    

  13. S. B. Malik, Some more properties of core partial order, Appl. Math. Comput. 221 (2013), 192-201.
    MathSciNet     CrossRef

  14. S. B. Malik, L. Rueda and N. Thome, Further properties on the core partial order and other matrix partial orders, Linear Multilinear Algebra 62 (2014), 1629-1648.
    MathSciNet     CrossRef

  15. J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra 63 (2015), 1707-1723.
    MathSciNet     CrossRef

  16. J. Marovt, One-sided sharp order in rings, J. Algebra Appl. 15 (2016), 1650161, 10 pp.
    MathSciNet     CrossRef

  17. J. Marovt, D. S. Rakić and D. S. Djordjević, Star, left-star, and right-star partial orders in Rickart *-rings, Linear Multilinear Algebra 63 (2015), 343-365.
    MathSciNet     CrossRef

  18. S. K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987), 17-37.
    MathSciNet     CrossRef

  19. S. K. Mitra, Matrix partial order through generalized inverses: unified theory, Linear Algebra Appl. 148 (1991), 237-263.
    MathSciNet     CrossRef

  20. S. K. Mitra, P. Bhimasankaram and S. B. Malik, Matrix partial orders, shorted operators and applications, World Scientific, Hackensack, 2010.
    MathSciNet     CrossRef

  21. M. Z. Nashed (ed.), Generalized inverses and applications, Academic Press, New York-London, 1976.
    MathSciNet    

  22. D. S. Rakić, Generalization of sharp and core partial order using annihilators, Banach J. Math. Anal. 9 (2015), 228-242.
    MathSciNet     CrossRef

  23. P. Šemrl, Automorphisms of B(𝓗) with respect to minus partial order, J. Math. Anal. Appl. 369 (2010), 205-213.
    MathSciNet     CrossRef

  24. B. Ungor, S. Halicioglu, A. Harmanci and J. Marovt, Partial orders on the power sets of Baer rings, J. Algebra Appl. 19 (2020), 2050011, 14 pp.
    MathSciNet     CrossRef
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