On the direct image of intersections in exact homological categories |
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Authors: | Dominique Bourn |
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Institution: | Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, Bat. H. Poincaré, 50 Rue F. Buisson BP 699, 62228 Calais, France |
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Abstract: | Given a regular epimorphism
f:X?Y in an exact homological category
C, and a pair
(U,V) of kernel subobjects of X, we show that the quotient
(f(U)∩f(V))/f(U∩V) is always abelian. When
C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair
(R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting
Y=X/T, this result says that the difference mapping determined by the inclusion
T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property. |
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Keywords: | 18G30 18B99 08A30 20J05 55U10 18C10 |
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