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 $(S,\oplus,\otimes)$ be a matrix semiring of max-plus algebra with the addition operation $\oplus$ and the multiplication operation $\otimes$, 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 \(\delta\) from \(S\) to itself that satisfies the axiom \(\delta(x \otimes y) = (\delta(x) \otimes y) \oplus (x \otimes \delta(y))\), for every \(x, y \in S\). 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 "$\dotplus$" and composition "$\circ$". We want to investigate the structure of $D$ over "$\dotplus$" and "$\circ$" operations. We show that \( D \) is not a semiring, but there exists a sub-semiring \( H \) \(\subseteq\) \( D \). Here, triple $(H,\oplus,\circ)$ is a semiring which is constructed from max-plus algebra.

Keywords

Main Subjects

References

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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

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History

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

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