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In universal algebra, homomorphisms are usually considered between algebras of the same similarity type. Different from that, the notion of a weak homomorphism, as introduced by E. Marczewski in 1961, does not depend on a signature, but only on the clones of term operations generated by the examined algebras. We generalize this idea by defining weak homomorphisms between F1 - and F2-algebras, where F1 and F2 denote not necessarily equal endofunctors of the category of sets. The aim is to show that, in many respects, weak homomorphisms behave very similarly to proper homomorphisms-without restricting the scope of considerations by the necessity of a common type. For instance, concerning a set F of Set -endofunctors that weakly preserve kernels, the class of all algebras of types from F equipped with the class of all weak homomorphisms between these algebras forms a category which admits a canonical factorization structure for morphisms. Furthermore, we treat two product constructions from which the notion of a weak homomorphism naturally arises.
Wydawca
Czasopismo
Rocznik
Tom
Strony
801--818
Opis fizyczny
Bibliogr. 10 poz.
Twórcy
autor
- Technical University Dresden Department Of Mathematics Institute Of Algebra 01062 Dresden, Germany, friedrich-martin.schneider@online.de
Bibliografia
- [1] J. Adámek, H. Herrlich, G. E. Srecker, Abstract and Concrete Categories: The Joy of Cats, Dover Publications (2009), 237–289.
- [2] K. Denecke, S. L. Wismath, Universal Algebra and Coalgebra, World Scientific Publishing (2009), 113–128.
- [3] K. Głazek, Weak homomorphisms of general algebras and related topics, Mathematics Seminar Notes 8 (1980), 1–36.
- [4] K. Głazek, Semigroups of weak endomorphisms of universal algebras, Proceedings of the International Symposium on the Semigroup Theory and its Related Fields, Kyoto (Aug. 30–Sept. 1), Japan, 1990, 85–102.
- [5] K. Głazek, Morphisms of general algebras without fixed fundamental operations, Contemp. Math. 184 (1991), 117–137.
- [6] K. Głazek, On weak automorphisms of some finite algebras, Contemp. Math. 131 (1992), 99–110.
- [7] V. Gorlov, R. Pöschel, Clones Closed with Respect to Permutation Groups or Transformation Semigroups, Dresden 1997, 1–30.
- [8] H. P. Gumm, Universal coalgebra, in: Thomas Ihringer, General Algebra, Berliner Studienreihe Math. 10 (2003), 155–208.
- [9] E. Marczewski, Independence and homomorphisms in abstract algebras, Fund. Math. 50 (1961), 45–61.
- [10] I. G. Rosenberg, A classification of universal algebras by infinitary relations, Algebra Universalis 1 (1972), 350–354.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA4-0033-0032