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SUMMARY:Embedding theorems for (regular) Mal'tsev categories - Pierre-Alai
 n Jacqmin (Université catholique de Louvain)
DTSTART:20170516T131500Z
DTEND:20170516T141500Z
UID:TALK72547@talks.cam.ac.uk
CONTACT:Tamara von Glehn
DESCRIPTION:Mal'tsev characterised [1] varieties of universal algebras for
  which the composition of congruences is commutative as varieties whose th
 eory contains a ternary operation p(x\,y\,z) satisfying p(x\,y\,y)=x and p
 (x\,x\,y)=y. This property of commutativity of the composition of equivale
 nce relations can be studied in the more general context of regular catego
 ries and gives rise to the notion of a regular Mal'tsev category [2]. More
 over\, a regular category is a Mal'tsev category if and only if every refl
 exive relation is an equivalence relation [2]. This last property makes se
 nse in any finitely complete category and enables us to extend the concept
  of Mal'tsev categories to this context [3]. Using the theory of approxima
 te Mal'tsev operations from [4]\, we present in this talk an essentially a
 lgebraic category M which is regular Mal'tsev and such that each small reg
 ular Mal'tsev category admits a conservative regular embedding into a powe
 r of M [5]. This gives a way to translate varietal proofs using elements a
 nd Mal'tsev operations into categorical proofs for regular Mal'tsev catego
 ries in general. Afterwards\, we also give an embedding theorem for weakly
  Mal'tsev and Mal'tsev categories in the category of partial Mal'tsev alge
 bras [6].\n\n[1] A. I. Mal'tsev\, On the general theory of algebraic syste
 ms\, Mat. Sbornik\, N.S. 35 (77) (1954)\, 3 -20"\n\n[2] A. Carboni\, J. La
 mbek and M.C. Pedicchio\, Diagram chasing in Mal'cev categories\, J. Pure 
 Appl. Algebra 69 (1990)\, 271- 284.\n\n[3] A. Carboni\, M.C. Pedicchio and
  N. Pirovano\, Internal graphs and internal groupoids in Mal'tsev categori
 es\, Canadian Math. Soc. Conf. Proc. 13 (1992)\, 97 -109.\n\n[4] D. Bourn 
 and Z. Janelidze\, Approximate Mal'tsev operations\, Theory and Appl. of C
 ateg. 21 No. 8 (2008)\, 152 -171.\n\n[5] P.-A. Jacqmin\, An embedding theo
 rem for regular Mal'tsev categories\, submitted to J. Pure Appl. Algebra.\
 n\n[6] P.-A. Jacqmin\, Partial Mal'tsev algebras and an embedding theorem 
 for (weakly) Mal'tsev categories\, submitted to Cah. Topol. Géom. Diff é
 r. Catég.
LOCATION:MR5\, Centre for Mathematical Sciences
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