Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T05:41:12.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Cogeneration of algebras in regular categories

Published online by Cambridge University Press:  17 April 2009

Jiří Adámek
Jiří Adámek
Affiliation:
Faculty of Electrical Engineering, České Vysoké Učení Technické v Praze, Czechoslovakia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A quotient algebra of a given algebra is said to be cogenerated by a quotient object if it is contained in the quotient object and is the biggest with this property. Triples T over a regular category are characterized which have the property that every quotient object of a T-algebra cogenerates some quotient algebra: these are precisely the right exact triples, preserving colimits of quotient chains. This improves a result of Michael Barr (J. Pure Appl. Algebra 4 (1974), 1–8) that every right exact, finitary triple has the investigated property. This result is related to categorical automata, since a triple has the above property iff triple machines admit a minimal realization of every behavior.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 355 - 370
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Adámek, Jiří, “Automata and categories: finiteness contra minimality”, Mathematical foundations of computer science 1975, 160–166 (Lecture Notes in Computer Science, 32. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[2]Adámek, Jiří, “Cogeneration and minimal realization”, Comment. Math. Univ. Carolinae (to appear).Google Scholar
[3]Adámek, Jiří, “Realization theory for automata in categories”, J. Pure Appl. Algebra (to appear).Google Scholar
[4]Adámek, Jiří, “On the cogeneration of algebras”, submitted.Google Scholar
[5]Arbib, M.A., Manes, E.G., “A categorist's view of automata and systems”, Category theory applied to computation and control, 51–64 (Lecture Notes in Computer Science, 25. Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[6]Barr, Michael, “Exact categories”, Exact categories and categories of sheaves, 1–120 (Lecture Notes in Mathematics, 236. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[7]Barr, Michael, “Right exact functors”, J. Pure Appl. Algebra 4 (1974), 1–8.CrossRefGoogle Scholar
[8]Grillet, Pierre Antoine, “Regular categories”, Exact categories and categories of sheaves, 121–222 (Lecture Notes in Mathematics, 236. Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[9]Hedrlín, Zdeněk and Pultr, Aleš, “On categorical embeddings of topological structures into algebraic”, Comment. Math. Univ. Carolinae 7 (1966), 377–400.Google Scholar
[10]Reiterman, Jan, “A left adjoint construction related to free triples”, J. Pure Appl. Algebra (to appear).Google Scholar
[11]Trnková, Věra, “On minimal realizations of behavior maps in categorial automata theory”, Comment. Math. Univ. Carolinae 15 (1974), 555–566.Google Scholar