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Green’s relation \(\mathcal{H}\) on the monoid of square matrices over a specific local ring. (English) Zbl 1509.20130

Summary: Let \(R\) be a commutative local ring whose maximal ideal is generated by a nilpotent element, and \(\mathrm{Mat}(n,R)\) be the multiplicative monoid of the square matrices of order \(n\) over \(R\). In this paper, we provide the construction of Green’s \(\mathcal{H} \)-equivalence classes in the multiplicative monoid \(\mathrm{Mat}(n,R)\). Then, we enumerate these classes in the special cases \(R=\mathbb{Z}/ p^d \mathbb{Z}\) and \(R= \mathbb{F}_q[x]/\langle x^d\rangle\).

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M10 General structure theory for semigroups
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

[1] Cao, Y., On the multiplicative monoid of \(n\timesn\) matrices over \(\Bbb Z_{p^m}\), Discrete Math.307 (2007) 3081-3096. · Zbl 1134.20065
[2] Cao, Y., On the multiplicativen monoid of \(n\timesn\) matrices over Artinian chain rings, Comm. Algebra38 (2010) 3404-3416. · Zbl 1227.20057
[3] Green, J. A., On the structure of semigroups, Ann. Math.54 (1951) 163-172. · Zbl 0043.25601
[4] Guterman, A., Johnson, M. and Kambites, M., Linear isomorphisms preserving Green’s relations for matrices over anti-negative semifields, Linear Algebra Appl.545 (2018) 1-14. · Zbl 1392.15040
[5] Levy, L. S. and Robson, J. C., Matrices and pairs of modules, J. Algebra29 (1974) 427-454. · Zbl 0282.16001
[6] Mary, X., On generalized inverses and Green’s relations, Linear Algebra Appl.434 (2011) 1836-1844. · Zbl 1219.15007
[7] Okniński, J., Semigroups of Matrices, , Vol. 6 (World Scientific, 1998). · Zbl 0911.20042
[8] Olshevsky, V., Strang, G. and Zhlobich, P., Green’s matrices, Linear Algebra Appl.432 (2010) 218-241. · Zbl 1195.15012
[9] Pei, H., Sun, L. and Zhai, H., Green’s relations for the variants of transformation semigroups preserving an equivalence relation, Comm. Algebra35 (2007) 1971-1986. · Zbl 1127.20044
[10] Petro, P., Green’s relations and minimal quasi-ideals in rings, Comm. Algebra30 (2006) 4677-4686. · Zbl 1051.16002
[11] Volkov, M. V., Silva, P. V. and Soares, F., Local finiteness for Green’s relations in semigroup varieties, Comm. Algebra46(2018) 4625-4653. · Zbl 1473.20062
[12] Yang, S. and Zhang, R., Green’s relations in the matrixsemigroup \(M_n(S)\), Linear Algebra Appl.222 (1995) 63-76. · Zbl 0830.20088
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