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On semigroups of endomorphisms of a chain with restricted range |
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| Authors: | Vítor H Fernandes Preeyanuch Honyam Teresa M Quinteiro Boorapa Singha |
| Institution: | 1. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica, Portugal 2. Centro de álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal 3. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand 4. Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, 1950-062, Lisboa, Portugal 5. School of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University, Chiang Mai, 50300, Thailand |
| Abstract: | Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\) . In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\) . Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks. |
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