Algebras with actions and automata
In the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The f...
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| Format: | Article |
| Language: | English |
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000076 |
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| author | W. Kühnel M. Pfender J. Meseguer I. Sols |
| author_facet | W. Kühnel M. Pfender J. Meseguer I. Sols |
| author_sort | W. Kühnel |
| collection | DOAJ |
| description | In the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The first section gives an axiomatic approach to algebraic structures relative to a base category B, slightly more powerful than that of monadic (tripleable) functors. In section 2 we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section 3 we treat the structures mentioned in the beginning as many-sorted algebras with fixed scalar or input object and show that they still have an algebraic (or monadic) forgetful functor (theorem 3.3) and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be a B-morphism), which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed natural numbers object has been studied by the authors in [23]. |
| format | Article |
| id | doaj-art-3fd825ff065d4bfcb53eea31d128caae |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3fd825ff065d4bfcb53eea31d128caae2025-02-03T05:52:54ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-0151618510.1155/S0161171282000076Algebras with actions and automataW. Kühnel0M. Pfender1J. Meseguer2I. Sols3Fachbereich Mathematik, TU Berlin, Str. d. 17. Juni 135, 1000, Berlin 12, GermanyFachbereich Mathematik, TU Berlin, Str. d. 17. Juni 135, 1000, Berlin 12, GermanyMathematics Department, University of California, Berkeley 94720, California, USAMathematics Department, University of California, Berkeley 94720, California, USAIn the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The first section gives an axiomatic approach to algebraic structures relative to a base category B, slightly more powerful than that of monadic (tripleable) functors. In section 2 we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section 3 we treat the structures mentioned in the beginning as many-sorted algebras with fixed scalar or input object and show that they still have an algebraic (or monadic) forgetful functor (theorem 3.3) and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be a B-morphism), which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed natural numbers object has been studied by the authors in [23].http://dx.doi.org/10.1155/S0161171282000076algebrasactionsautomataalgebraic functor. |
| spellingShingle | W. Kühnel M. Pfender J. Meseguer I. Sols Algebras with actions and automata International Journal of Mathematics and Mathematical Sciences algebras actions automata algebraic functor. |
| title | Algebras with actions and automata |
| title_full | Algebras with actions and automata |
| title_fullStr | Algebras with actions and automata |
| title_full_unstemmed | Algebras with actions and automata |
| title_short | Algebras with actions and automata |
| title_sort | algebras with actions and automata |
| topic | algebras actions automata algebraic functor. |
| url | http://dx.doi.org/10.1155/S0161171282000076 |
| work_keys_str_mv | AT wkuhnel algebraswithactionsandautomata AT mpfender algebraswithactionsandautomata AT jmeseguer algebraswithactionsandautomata AT isols algebraswithactionsandautomata |