Algebras with actions and automata



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Kühnel, Wolfgang and Pfender, Michael and Meseguer Guaita, José and Sols, Ignacio (1982) Algebras with actions and automata. International Journal of Mathematics and Mathematical Sciences, 5 (1). pp. 61-85. ISSN 0161-1712

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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].

Item Type:Article
Uncontrolled Keywords:Algebras, actions, automata, algebraic functor.
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:21494
Deposited On:23 May 2013 15:21
Last Modified:27 Jul 2018 08:25

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