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Strongly regular matrices and simple image set in interval max-plus algebra. (English) Zbl 1408.15011

Summary: Suppose that \(\mathbb R\) is the set of real numbers and \(\mathbb R_{\varepsilon} = \mathbb R \cup \{\varepsilon \}\) with \(\varepsilon = -\infty\) Max-plus algebra is the set \(\mathbb R_{\varepsilon}\) that is equipped with two operations, maximum and addition. Matrices over max-plus algebra are matrices whose elements belong to \(\mathbb R_{\varepsilon}\). The set \(I(\mathbb R)_{\varepsilon} = \{ x= [\underline x, \bar x] \underline x, \bar x \in \mathbb R, \varepsilon < \underline x \leq \bar x \} \cup \{\varepsilon\}\) with \(\varepsilon = [\varepsilon,\varepsilon]\) is equipped with maximum and addition operations is known as interval max-plus algebra. Matrices over interval max-plus algebra are matrices whose elements belong to \(I(\mathbb R_{\varepsilon})\). The research about image set, strongly regular matrix and simple image set in max-plus algebra have been done. In this paper, we investigate image set, strongly regular matrix and simple image set in interval max-plus algebra.

MSC:

15A80 Max-plus and related algebras
15A15 Determinants, permanents, traces, other special matrix functions
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A30 Algebraic systems of matrices
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