Yang, Shangjun; Barker, George P. Generalized Green’s relations. (English) Zbl 0791.20072 Czech. Math. J. 42, No. 2, 211-224 (1992). 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. Reviewer: J.A.Hildebrant (Baton Rouge) 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 Keywords:Green’s relations; action; monoids; stable combine; actions with compact monoids; matrix combines PDFBibTeX XMLCite \textit{S. Yang} and \textit{G. P. Barker}, Czech. Math. J. 42, No. 2, 211--224 (1992; Zbl 0791.20072) Full Text: EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.