caa a22 4500 445378689 CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10485-009-9192-5 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9192-5 Split Short Five Lemma for Clots and Subtractive Categories [Elektronische Daten] [Zurab Janelidze, Aldo Ursini] It is well-known that a pointed variety is classically ideal determined (or "BIT speciale”) if and only if it satisfies the split short five lemma (i.e.if and only if it is a protomodular category in the sense of D.Bourn). A much weaker property than being classically ideal determined is "subtractivity”, defined as follows: a variety with a constant 0 is said to be subtractive if its theory contains a binary term s satisfying s(x,x) = 0 and s(x,0) = x. In the case of a pointed variety (i.e.when 0 is the unique constant), this condition can be reformulated purely categorically (as many other similar term conditions), which gives rise to the notion of a subtractive category. In the present paper we show that in a certain general categorical context subtractivity is equivalent to a special restriction of the split short five lemma to the class of clots, i.e.monomorphisms that are pullbacks of reflexive relations R→Y×Y along product injections (1Y,0): Y→Y×Y. Springer Science+Business Media B.V., 2009 Subtractive category nationallicence Subtractive variety nationallicence Split short five lemma nationallicence Clot nationallicence Classically ideal determined variety nationallicence 0-coherent variety nationallicence Protomodular category nationallicence Semi-abelian category nationallicence Homological category nationallicence Janelidze Zurab Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, 7602, Matieland, South Africa aut Ursini Aldo Dipartimento di Scienze Matematiche ed Informatiche, Universita' di Siena, Pian dei Mantellini 44, 53100, Siena, Italy aut Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 233-255 0927-2852 19:1<233 2011 19 10485 https://doi.org/10.1007/s10485-009-9192-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9192-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Janelidze Zurab Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, 7602, Matieland, South Africa aut NATIONALLICENCE 700 1- Ursini Aldo Dipartimento di Scienze Matematiche ed Informatiche, Universita' di Siena, Pian dei Mantellini 44, 53100, Siena, Italy aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 233-255 0927-2852 19:1<233 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer