Abstract
We study relations that are induced by the core-EP decomposition. We revisit the core-EP preorder and extend the concepts of the core-minus, the c-minus, and the c-star partial orders from the set of all $n\times n$ complex matrices to the set of all core-EP invertible elements in a $\ast$-ring. Several characterizations of these relations are presented and thus some known results are generalized.
References
- O. M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra, 58(6) (2010), 681-697.
- D. S. Djordjevic, D. S. Rakic and J. Marovt, Minus partial order in Rickart rings, Publ. Math. Debrecen, 87(3/4) (2015), 291-305.
- G. Dolinar, B. Kuzma and J. Marovt, A note on partial orders of Hartwig, Mitsch, and Semrl, Appl. Math. Comput., 270 (2015), 711-713.
- G. Dolinar, B. Kuzma, J. Marovt and B. Ungor, Properties of core-EP order in rings with involution, Front. Math. China, 14(4) (2019), 715-736.
- G. Dolinar, J. Marovt and D. Mosic, On EP-nilpotent decomposition in $\ast$-rings, Publ. Math. Debrecen, 100(3/4) (2022), 405-426.
- M. P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc., 84(1) (1978), 139-141.
- Y. Gao and J. Chen, Pseudo core inverses in rings with involution, Comm. Algebra, 46(1) (2018), 38-50.
- Y. Gao, J. Chen and Y. Ke, *-DMP elements in $\ast $-semigroups and $\ast$-rings, Filomat, 32 (2018), 3073-3085.
- R. E. Hartwig, How to partially order regular elements,Math. Japon., 25 (1980), 1-13.
- R. E. Hartwig and J. Luh, A note on the group structure on unit regular ring elements, Pacific J. Math., 71 (1977), 449-461.
- A. Herrero and N. Thome, Sharp partial order and linear autonomous systems, Appl. Math. Comput., 366 (2020), 124736 (11 pp).
- P. M. Higgins, Techniques of Semigroup Theory, Oxford University Press, New York, 1992.
- Q. Ling, X. Hua, H. Wang and S. Liu, The C-N-star, s-star and c-minus partial orders, J. Math. Inequal., 19(1) (2025), 177-194.
- K. Manjunatha Prasad and K. S. Mohana, Core-EP inverse, Linear Multilinear Algebra, 62(6) (2014), 792-802.
- J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra, 63(9) (2015), 1707-1723.
- J. Marovt, On star, sharp, core, and minus partial orders in Rickart rings, Banach J. Math. Anal., 10(3) (2016), 495-508.
- J. Marovt and D. Mosic, On some orders in $\ast $-rings based on the core-EP decomposition, J. Algebra Appl., 21(1) (2022), 2250010 (24 pp).
- S. K. Mitra, P. Bhimasankaram and S. B. Malik, Matrix Partial Orders, Shorted Operators and Applications, World Scientific, Hackensack, 2010.
- H. Mitsch, A natural partial order for semigroups, Proc. Amer. Math. Soc., 97(3) (1986), 384-388.
- D. Mosic, Weighted core-EP inverse and weighted core-EP pre-orders in a $C^{\ast}$-algebra, J. Aust. Math. Soc., 111(1) (2021), 76-110.
- K. S. S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc., 23 (1980), 249-260.
- D. S. Rakic and D. S. Djordjevic, Star, sharp, core and dual core partial order in rings with involution, Appl. Math. Comput., 259 (2015), 800-818.
- C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.
- P. Semrl, Automorphism of $\mathcal{B}(H)$ with respect to minus partial order, J. Math. Anal. Appl., 369 (2010), 205-213.
- S. Xu, J. Chen and X. Zhang, New characterizations for core inverses in rings with involution, Front. Math. China, 12(1) (2017), 231-246.
- H. Wang, Core-EP decomposition and its applications, Linear Algebra Appl., 508 (2016), 289-300.
- D. Zhang, L. Ma and D. Mosic, Representations of the Moore-Penrose inverse of a $2\times 2$ block matrix based on Schur complements, Appl. Math. Comput., 512 (2026), 129776 (10 pp).
- D. Zhang, Y. Zhao and D. Mosic, The generalized Drazin inverse of the sum of two elements in a Banach algebra, J. Comput. Appl. Math., 470 (2025), 116701 (12 pp).
Abstract
References
- O. M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra, 58(6) (2010), 681-697.
- D. S. Djordjevic, D. S. Rakic and J. Marovt, Minus partial order in Rickart rings, Publ. Math. Debrecen, 87(3/4) (2015), 291-305.
- G. Dolinar, B. Kuzma and J. Marovt, A note on partial orders of Hartwig, Mitsch, and Semrl, Appl. Math. Comput., 270 (2015), 711-713.
- G. Dolinar, B. Kuzma, J. Marovt and B. Ungor, Properties of core-EP order in rings with involution, Front. Math. China, 14(4) (2019), 715-736.
- G. Dolinar, J. Marovt and D. Mosic, On EP-nilpotent decomposition in $\ast$-rings, Publ. Math. Debrecen, 100(3/4) (2022), 405-426.
- M. P. Drazin, Natural structures on semigroups with involution, Bull. Amer. Math. Soc., 84(1) (1978), 139-141.
- Y. Gao and J. Chen, Pseudo core inverses in rings with involution, Comm. Algebra, 46(1) (2018), 38-50.
- Y. Gao, J. Chen and Y. Ke, *-DMP elements in $\ast $-semigroups and $\ast$-rings, Filomat, 32 (2018), 3073-3085.
- R. E. Hartwig, How to partially order regular elements,Math. Japon., 25 (1980), 1-13.
- R. E. Hartwig and J. Luh, A note on the group structure on unit regular ring elements, Pacific J. Math., 71 (1977), 449-461.
- A. Herrero and N. Thome, Sharp partial order and linear autonomous systems, Appl. Math. Comput., 366 (2020), 124736 (11 pp).
- P. M. Higgins, Techniques of Semigroup Theory, Oxford University Press, New York, 1992.
- Q. Ling, X. Hua, H. Wang and S. Liu, The C-N-star, s-star and c-minus partial orders, J. Math. Inequal., 19(1) (2025), 177-194.
- K. Manjunatha Prasad and K. S. Mohana, Core-EP inverse, Linear Multilinear Algebra, 62(6) (2014), 792-802.
- J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra, 63(9) (2015), 1707-1723.
- J. Marovt, On star, sharp, core, and minus partial orders in Rickart rings, Banach J. Math. Anal., 10(3) (2016), 495-508.
- J. Marovt and D. Mosic, On some orders in $\ast $-rings based on the core-EP decomposition, J. Algebra Appl., 21(1) (2022), 2250010 (24 pp).
- S. K. Mitra, P. Bhimasankaram and S. B. Malik, Matrix Partial Orders, Shorted Operators and Applications, World Scientific, Hackensack, 2010.
- H. Mitsch, A natural partial order for semigroups, Proc. Amer. Math. Soc., 97(3) (1986), 384-388.
- D. Mosic, Weighted core-EP inverse and weighted core-EP pre-orders in a $C^{\ast}$-algebra, J. Aust. Math. Soc., 111(1) (2021), 76-110.
- K. S. S. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc., 23 (1980), 249-260.
- D. S. Rakic and D. S. Djordjevic, Star, sharp, core and dual core partial order in rings with involution, Appl. Math. Comput., 259 (2015), 800-818.
- C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.
- P. Semrl, Automorphism of $\mathcal{B}(H)$ with respect to minus partial order, J. Math. Anal. Appl., 369 (2010), 205-213.
- S. Xu, J. Chen and X. Zhang, New characterizations for core inverses in rings with involution, Front. Math. China, 12(1) (2017), 231-246.
- H. Wang, Core-EP decomposition and its applications, Linear Algebra Appl., 508 (2016), 289-300.
- D. Zhang, L. Ma and D. Mosic, Representations of the Moore-Penrose inverse of a $2\times 2$ block matrix based on Schur complements, Appl. Math. Comput., 512 (2026), 129776 (10 pp).
- D. Zhang, Y. Zhao and D. Mosic, The generalized Drazin inverse of the sum of two elements in a Banach algebra, J. Comput. Appl. Math., 470 (2025), 116701 (12 pp).
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Research Article |
| Authors | |
| Submission Date | September 8, 2025 |
| Acceptance Date | January 11, 2026 |
| Early Pub Date | February 18, 2026 |
| Publication Date | February 18, 2026 |
| DOI | https://doi.org/10.24330/ieja.1892320 |
| IZ | https://izlik.org/JA38ZN28EB |
| Published in Issue | Year 2026 Issue: Advanced Online Publication |
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