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Minimal Generators in Varieties

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Abstract

Given a variety V, the free V-algebra F 1 on one generator represents the canonical underlying functor from V to the category Set of sets. Hence one might ask, whether F 1 is, in some sense, a `canonical' generator of V. To make this question precise the notion of `minimal varietal generator' is introduced. It is shown that in many (though not all) varieties F 1 is a generator of this kind, and often even the unique one. The question whether every variety has a (unique) minimal varietal generator, remains on open problem.

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References

  1. Adámek, J. and Rosický, J.: Locally Presentable and Accessible Categories, Cambridge Univ. Press, Cambridge, 1994.

    Google Scholar 

  2. Barr, M. and Wells, C.: Toposes, Triples and Theories, Springer, 1984.

  3. Börger, R.: Morita equivalence for monads, Mathematik-Arbeitspapiere 47 (1997), 67–76.

    Google Scholar 

  4. Herrlich, H. and Strecker, G. E.: Category Theory, 2nd edn, Heldermann Verlag, 1979.

  5. Isbell, J. R.: Subobjects, adequacy, completeness, and categories of algebras, Rozprawy Matem. 36 (1964).

  6. Jacobson, N.: Basic Algebra, Vol. II, Freeman, 1980.

  7. MacLane, S.: Duality for groups, Bull. Amer. Math. Soc. 56 (1950), 485–516.

    Google Scholar 

  8. Porst, H.-E.: On categories of monoid actions, Quaestiones Math. 10 (1987), 391–411.

    Google Scholar 

  9. Porst, H.-E.: Equivalence of varieties, Preprint, 1998.

  10. Semadeni, Z.: Banach Spaces of Continuous Functions, Warszaw, 1971.

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Porst, HE. Minimal Generators in Varieties. Applied Categorical Structures 8, 519–525 (2000). https://doi.org/10.1023/A:1008778128355

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  • DOI: https://doi.org/10.1023/A:1008778128355

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