Abstract
H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality \({{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}\) for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove \({{\mathcal {HS}} \neq {\mathcal {SH}}}\) for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adamék J., Trnková V.: Automata and Algebras in a Category. Kluwer Publishing Company, Dordrecht (1990)
Brengos T., Trnková V.: The \({{\mathcal {HS}} = {\mathcal {SH}}}\) problem for coalgebras. Algebra Universalis 63, 283–302 (2010)
Gumm H.P.: Elements of the general theory of coalgebras. LUATCS 99, Rand Afrikaans University, Johannesburg (1999)
Gumm H.P., Schröder T.: Types and coalgebraic structure. Algebra Universalis 53, 229–252 (2005)
Gumm, H. P.: From T-coalgebras to filter structures and transtion systems. In: Algebra and Coalgebra in Computer Science (CALCO 2005). Lecture Notes in Computer Science, vol. 3629, pp. 194–212. Springer (2005).
Jech, T.: Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics (2002).
Pultr, A., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland and Academia Praque, Prague (1980).
Trnková V.: Some properties of Set-endofunctors. Commentationes Mathematicae Universitatis Carolinae 10, 323–259 (1969)
Trnková V.: On descriptive classification of Set-functors I. Commentationes Mathematicae Universitatis Carolinae 12, 143–174 (1971)
Vopěnka P., Pultr A., Hedrlín Z.: A rigid relation exists on any set. Commentationes Mathematicae Universitatis Carlinae 6, 149–155 (1965)
Zmrzlina, A.: Too Many Functors - A continuation of “The Emergence of Functors”. In: Categorical Perpectives. Trends in Mathematics, pp. 47–62. Birkhauser Verlag (2001)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by H. P. Gumm.
Research supported by the Warsaw University of Technology under grant number 504G/1120/0054/000.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Brengos, T. Rigidity of unary algebras and its application to the \({\mathcal {HS} = \mathcal {SH}}\) problem. Algebra Univers. 65, 73–89 (2011). https://doi.org/10.1007/s00012-011-0118-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-011-0118-3