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Zbl 0778.18004
Henry, C.
Permanence properties and existence of minimal realizations of many-sorted algebras. (Propriétés de permanence et existence de réalisations minimales pour les algèbres multisortes.) (French)
[J] Diagrammes 27, CH 1-CH 53 (1992). ISSN 0224-3911

The paper consists of an introduction and two parts. In the introduction basic facts concerning the theory of many-sorted algebraic theories are presented. In Part 1 sectionable homomorphisms and retractable homomorphisms between many-sorted theories are introduced and examined. The definitions are too lengthy to be quoted here. Let $H:T\to T'$ be a homomorphism of many-sorted theories. It is known that the induced functor $\text{Alg}(H):\text{Alg}(T')\to\text{Alg}(T)$ between the categories of corresponding algebras has a left adjoint. Let $e$ be the unit of this adjunction. The author proves that (a) if $H$ is sectionable [retractable], then all the components of $e$ are epimorphisms [monomorphisms], (b) $H$ is sectionable and retractable if and only if all the components of $e$ are isomorphisms (the property of permanence). In Part 2 the author follows the ideas of J. A. Goguen and J. Meseguer and proves some theorems concerning the existence of minimal realizations for many-sorted theories. Two examples are discussed: automata, and actions of groups on pointed sets.
[A.Wiweger (Warszawa)]

MSC 2000:: *18C10 Algebraic theories, etc.
08A30 Subalgebras of general algebraic systems
08B20 Free algebras in varieties of algebras
18A40 Adjoint functors
08A70 Appl. of universal algebra in computer science
68Q70 Algebraic theory of automata

Keywords: adjoint functor; many-sorted algebra; many-sorted theory; minimal realization; unit of an adjunction

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