# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
  D.Brunker,Topics in the algebra of axiomatic infinite sums, University of Waterloo Ph.D. thesis, 1980.  I. Fleischer,Congruence extension from a semilattice to the freely generated distributive lattice, Czech. Math. J.32 (107) (1982) 623-626. · Zbl 0512.06007  I.Fleischer,Co-compositive partial algebras.  H. Herrlich andG. E. Strecker,Category Theory, Boston, Allyn & Bacon 1973. · Zbl 0265.18001  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  F. E. Linton,Autonomous categories and duality of functors, J. Alg.2 (1965) 315-349. · Zbl 0166.27503  F. E. J. Linton,Autonomous equational categories, J. Math. Mech.15 (1966) 637-642. · Zbl 0146.25104  H. M. Macneille,Partially ordered sets, Trans. Amer. Math. Soc.42 (1937) 416-460. · Zbl 0017.33904  J.Slomi?ski,A theory of extensions of quasi-algebras to algebras, Rozprawy Math.40 (1964).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.