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Sur les Opérations Partielles Implicites et Leur Relation Avec la Surjectivité Des Épimorphismes

Published online by Cambridge University Press:  20 November 2018

Michel Hébert
Michel Hébert*
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, Québec, G1K 7P4, email: [email protected]
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Abstract

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Let K be a category of structures with all its homomorphisms, Uα K → Set the α-th power of its forgetful functor U An α-ary implicit partial operation (O.P.I.) in K is a diagram of natural transformations and functors. We first study various properties which O P I's can have, as maximality, definability and closure under products or equalizers. Revisiting various concepts and results of Isbell, Linton, Bacsich and Herrera, we note, among other things, that the dominion (resp. the stable dominion) of a subset of a structure K is its closure under O P I's (resp. under equalizer-closed O P I's), and we show that in a variety, all epis are surjective (resp. all monos are regular) iff all limit-closed (resp. product-closed) O P I's are restrictions of total implicit operations.

Keywords

08A5508A4018A2003C40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 554 - 575
Copyright
Copyright © Canadian Mathematical Society 1993

References

Références

1. Almeida, J., Equations for pseudovarietie s, LITP 88, LNCS 386, (1989).Google Scholar
2. Bacsich, P. D., Model theory ofepimorphisms, Canad. Math. Bull.l7(1974), 471–477.Google Scholar
3. Burmeister, P. and Reichel, H., Introduction to theory and applications of partial algebras, Akademie-Verlag, Berlin, 1984.Google Scholar
4. Burns, S. and Sankappanavar, H. P.,A course in Universal Algebra, Springer-Verlag, New York, 1981.Google Scholar
5. Cassidy, C., Hébert, M. and Kelly, G. M., Reflexive subcategories, localizations and factorization systems, J. Austr. Math. Soc. 38(1985), 287–329.Google Scholar
6. Castellini, G., Compact objects, surjectivity ofepimorphismsandcompactifications,Cah. Top. Géom. Diff. Cat. (1)31(1990).Google Scholar
7. Chang, C. C. and Keisler, H. J., Model Theory, Springer-Verlag, Amsterdam, 1973.Google Scholar
8. Diers, Y., Familles universelles de morphismes, Ann. Soc. Math. Bruxelles 93(1979), 175–195.Google Scholar
9. Hébert, M., Characterizations of axiomatic categories of models canonically isomorphic to (quasi-) varieties, Canad. Math. Bull. 31(1988), 287–300.Google Scholar
10. Hébert, M., Sur le rang et la définissabilité des opérations implicites dans les classes d'algèbres, Coll. Math. 58(1990), 175–188.Google Scholar
11. Hébert, M., Sur les algèbres libres pour les classes axiomatiques, Ann. Sci. Math. Que. 13(1989), 39–47.Google Scholar
12. Herrera, J., Les théories convexes de Horn, CR. Acad. Se. Paris (A) 287(1978), 593–594.Google Scholar
13. Herrlich, H. and Strecker, G. E., Category Theory, Helderman-Verlag, Berlin, 1979.Google Scholar
14. Hodges, W., Functorial implicit reducibility, Fund. Math. 108(1980), 77–81.Google Scholar
15. Isbell, J. R., Epimorphisms and dominions. In: Proc. Conf. Categ. Alg., La Jolla, (1965), Springer-Verlag, New York, (1966), 232–246.Google Scholar
16. Isbell, J. R., Functorial implicit operations, Israel J. Math. 15(1973), 185–187.Google Scholar
17. Kelly, G. M., Monomorphisms, epimorphisms and pull-backs, J. Austr. Math. Soc. 9(1969), 124–142.Google Scholar
18. Kennison, J. F. and Guldenhuys, D., Equational completion, model-induced triples and pro-objects, J. Pure Appl. Alg. 1(1971), 317–346.Google Scholar
19. Kiss, E. W., Marki, L., Prohle, P. and Tholen, W., Categorical algebraic properties. A compendium…, Studia Sc. Math. Hung. 18(1983), 79–141.Google Scholar
20. Lawvere, F. W., Some algebraic problems in the context of functorial semantics of algebraic theories, LNM 61(1968),41–61.Google Scholar
21. J. Linton, F. E., Some aspects of equational categories. In: Proc. Conf. Categ. Alg., La Jolla, (1965), Springer-Verlag, New York, (1966), 84–94.Google Scholar
22. Manca, V., Salibra, A. and Scollo, G., Equational type logic, Theor. Comp. Sc. 77(1990), 131–159.Google Scholar
23. Picavet, G., Pureté, rigidité et morphismes entiers, Trans. Amer. Math. Soc. 323(1991), 283–313.Google Scholar
24. Tarlecki, A., On the existence of free models in abstract algebraic institutions, Theor. Comp. Sc. 37(1985), 269–304.Google Scholar
25. Volger, H., Preservation theorems for limits of structures and global sections of structures, Math. Z. 166 (1979), 27–53.Google Scholar
26. Volger, H., The model theory of disjunctive logic programs, Comp. Sci. Logic 89, LNCS 440(1990).Google Scholar