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Preservation of bi-endomorphic composition by categorical reflection. (English) Zbl 0674.08006
Given a concrete category $${\mathcal C}$$ over Set (with forgetful functor U), a bi-endomorphic composition on a $${\mathcal C}$$-object A is a binary operation on the underlying set U(A) such that both exponential transposes $$U(A) \rightrightarrows [U(A),U(A)]$$ factor through the hom- set $${\mathcal C}(A,A)$$. If $$\bar {\mathcal C}$$ is a (full) reflective subcategory of $${\mathcal C}$$, one may ask whether the two compositions $$U(A) \rightrightarrows {\mathcal C}(A,A)\to{\mathcal C}(\bar A,\bar A)$$ factor through the underlying map of the universal morphism $$A\to \bar A$$, and if they do, whether the resulting functions $$U(\bar A) \rightrightarrows {\mathcal C}(\bar A,\bar A)\to [U(\bar A),U(\bar A)]$$ are in fact the exponential transposes of some bi-endomorphic composition on $$\bar A.$$ The author uses this approach to compare reflections in categories $${\mathcal C}$$ and $${\mathcal C}'$$ of partial algebras and domain-preserving homomorphisms, where the $${\mathcal C}'$$-objects are $${\mathcal C}$$-objects with an additional total bi-endomorphic composition. In good cases, the reflection in $${\mathcal C}'$$ reduces to the reflection in $${\mathcal C}$$, thus eliminating the need for a separate construction. As an example, the relatively free $$\Sigma$$- ring can simply be obtained by forming the corresponding relatively free $$\Sigma$$-group.
Reviewer: J.Koslowski

##### MSC:
 08C05 Categories of algebras 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
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##### References:
 [1] D.Brunker,Topics in the algebra of axiomatic infinite sums, University of Waterloo Ph.D. thesis, 1980. [2] I. Fleischer,Congruence extension from a semilattice to the freely generated distributive lattice, Czech. Math. J.32 (107) (1982) 623-626. · Zbl 0512.06007 [3] I.Fleischer,Co-compositive partial algebras. [4] H. Herrlich andG. E. Strecker,Category Theory, Boston, Allyn & Bacon 1973. · Zbl 0265.18001 [5] D.Higgs,Axiomatic infinite sums ? an algebraic approach to integration theory, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces: Deducated to B. J. Pettis, A.M.S. Series in Contemporary Mathematics, 1980. · Zbl 0554.28011 [6] F. E. Linton,Autonomous categories and duality of functors, J. Alg.2 (1965) 315-349. · Zbl 0166.27503 [7] F. E. J. Linton,Autonomous equational categories, J. Math. Mech.15 (1966) 637-642. · Zbl 0146.25104 [8] H. M. Macneille,Partially ordered sets, Trans. Amer. Math. Soc.42 (1937) 416-460. · Zbl 0017.33904 [9] J.Slomi?ski,A theory of extensions of quasi-algebras to algebras, Rozprawy Math.40 (1964).
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