Derivations on the matrix semirings of max-plus algebra

Document Type : Special Issue: First Joint IIIMT-Algebra Forum Conference 2023

Authors

Department of Mathematics, Universitas Diponegoro, Semarang, Indonesia

Abstract

Let (S,⊕,⊗) be a matrix semiring of max-plus algebra with the addition operation ⊕ and the multiplication operation ⊗, where the set S consists of matrices constructed from real numbers together with the element negative infinity. A derivation on the semiring S is an additive mapping δ from S to itself that satisfies the axiom δ(x⊗y)=(δ(x)⊗y)⊕(x⊗δ(y)), for every x,y∈S. From S we construct all of semiring derivations of S are denoted by D. On the set D, we defined two binary operations, i.e., addition "∔" and composition "∘". We want to investigate the structure of D over "∔" and "∘" operations. We show that D is not a semiring, but there exists a sub-semiring H ⊆ D. Here, triple (H,⊕,∘) is a semiring which is constructed from max-plus algebra.

Keywords

Main Subjects

References

[1] Ansari, A. Z., Shujat, F., & Fallatah, A. (2023). Generalized di erential identities on prime rings and algebras. AIMS Mathematics, 8(10), 22758-22765. https://doi.org/10.3934/math.20231159
[2] Ansari, A. Z., & Shujat, F. (2023). Jordan -derivations on standard operator algebras. Filomat, 37(1), 37-41. https://doi.org/10.2298/FIL2301037A
[3] Ashraf, M., Ali, S., & Haetinger, C. (2006). On derivations in rings and their applications. The Aligarh Bulletin of Mathematics, 25(2), 79-107.
[4] Baccelli, F., Cohen, G., Olsder, G. L., & Quadrat, J. P. (1994). Synchronization and linearity: An algebra for discrete event systems. Journal of the Operational Research Society, 45(1), 15. https://doi.org/10.1057/JORS.1994.15
[5] Dimitrov, S. (2017). Derivations on semirings. AIP Conference Proceedings, 1910(1), 060011. https://doi.org/10.1063/1.5014005
[6] Fraleigh, J. B. (1999). A  rst course in abstract algebra (7th ed.). Addison Wesley Publishing Company.
[7] Golan, J. S. (1999). Semirings and their applications. Springer Dordrecht. https://doi.org/10.1007/978-94-015-9333-5
[8] Heidergott, B., Olsder, G. J., & van der Woude, J. (2006). Max plus at work: Modeling and analysis of synchronized systems: A course on max-plus algebra and its applications. Princeton University Press. https://doi.org/10.1515/9781400865239
[9] Mceneaney, W. (2006). Max-plus methods for nonlinear control and estimation. Boston: Birkhauser. https://doi.org/10.1007/0-8176-4453-9
[10] Olsder, G. J., & Roos, C. (1988). Cramer and Cayley-Hamilton in the max algebra. Linear Algebra and its Applications, 101, 87-108. https://doi.org/10.1016/0024-3795(88)90145-0
[11] Schutter, B. D., & Moor, B. D. (1997). A note on the characteristic equation in the max-plus algebra. Linear Algebra and its Applications, 261(1), 237-250. https://doi.org/10.1016/S0024-3795(96)00407-7
[12] Vandiver, H. S. (1934). Note on a simple type of algebra in which the cancellation law of addition does not hold. Bulletin of the American Mathematical Society, 40(12), 914-920. https://doi.org/10.1090/S0002-9904-1934-06003-8
 
Journal of Mahani Mathematical Research
Volume 13, Issue 5 - Serial Number 30
Special Issue: First Joint IIIMT-Algebra Forum Conference 2023
December 2024
Pages 51-63

Files

History

  • Receive Date: 30 April 2024
  • Revise Date: 10 November 2024
  • Accept Date: 25 December 2024

Share

How to cite

Statistics