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Generalized Green’s relations. (English) Zbl 0791.20072

The authors extend the concept of Green’s relations on a semigroup to the action of one or more monoids on another. If \(U\) and \(V\) act on \(T\) and \(u(tv) = (ut)v\), for \(u\in U\), \(v \in V\), and \(t\in T\), then \(T\) is called a \(U\)-\(V\) combine. A corresponding notion of a stable combine is consequential, the subsequent conclusion that \({\mathcal D} = {\mathcal J}\) in a stable combine, and that actions with compact monoids result in a stable combine. A further discussion of Green’s relation in this setting follows along with an extensive study of Green’s relations for matrix combines.

MSC:

20M10 General structure theory for semigroups
20M15 Mappings of semigroups
20M50 Connections of semigroups with homological algebra and category theory
15A30 Algebraic systems of matrices
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References:

[1] G. P. Barker, Yang Shangjun: Structure of \(\mathcal F\)-classes in the semigroup of nonnegative matrices. · Zbl 1075.20517
[2] A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Academic Press, Inc. New York. 1979. · Zbl 0484.15016
[3] J. H. Carruth J. A. Hildebrandt, R. J. Koch: T\?e Theory of Topological Semigroups. Marcel Dekker, Inc. New York. 1983. · Zbl 0515.22003
[4] A. H. Clifford, G. B. Preston: The Algebraic Theory of Semigroups, I. Math. Surveys No. 7, American Math. Soc. Providence, RI 1961. · Zbl 0111.03403
[5] D. J. Hartfiel C. J. Maxson, R. J. Plemmons: A Note on Green’s relations on the semigroup \(\mathcal N\). Proc. Amer. Math. Soc. 60, 11-15) · Zbl 0351.20037 · doi:10.2307/2041100
[6] N. Jacobson: Basic Algebra. I. W. H. Freeman and Co. San Francisco, CA. 1974. · Zbl 0284.16001
[7] [7l C. E. Robinson, Jr.: Green’s relations for substochastic matrices. Linear Algebra and its Applications, 80, 39-53 (1986). · Zbl 0599.15012 · doi:10.1016/0024-3795(86)90276-4
[8] Yang Shangjun: Structure of \(\mathcal K\)-classes in the semigroup of nonnegative marices. Linear Algebra and its Applications, 60, 91-111 (1984). · Zbl 0542.15005 · doi:10.1016/0024-3795(84)90073-9
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