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Rigidity of unary algebras and its application to the \({\mathcal {HS} = \mathcal {SH}}\) problem. (English) Zbl 1221.08004
H. P. Gumm and T. Schröder [Algebra Univers. 53, No. 2–3, 229–252 (2005; Zbl 1086.08002)] showed that if a functor \(T:\) {Set} \(\to\) {Set} preserves preimages, then \({\mathcal{H}}{\mathcal{S}}(K)={\mathcal{S}}{\mathcal{H}}(K)\) is valid for any class \(K\) of \(T\)-algebras. The present author applies a construction of rigid unary algebras to prove that \({\mathcal{H}}{\mathcal{S}}\neq{\mathcal{S}}{\mathcal{H}}\) for a class of {Set}-endofunctors not preserving non-empty preimages. For related results cf. also papers by T. Brengos and V. Trnková [Algebra Univers. 63, No. 2–3, 283–301 (2010; Zbl 1220.03011)] and V. Trnková [Commentat. Math. Univ. Carol. 10, 323–352 (1969; Zbl 0183.30401); ibid. 12, 143–174 (1971; Zbl 0232.18004)].
MSC:
08A60 Unary algebras
03G30 Categorical logic, topoi
08A70 Applications of universal algebra in computer science
18A22 Special properties of functors (faithful, full, etc.)
18B05 Categories of sets, characterizations
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[1] Adamék J., Trnková V.: Automata and Algebras in a Category. Kluwer Publishing Company, Dordrecht (1990)
[2] Brengos T., Trnková V.: The $${{\(\backslash\)mathcal {HS}} = {\(\backslash\)mathcal {SH}}}$$ problem for coalgebras. Algebra Universalis 63, 283–302 (2010) · Zbl 1220.03011
[3] Gumm H.P.: Elements of the general theory of coalgebras. LUATCS 99, Rand Afrikaans University, Johannesburg (1999)
[4] Gumm H.P., Schröder T.: Types and coalgebraic structure. Algebra Universalis 53, 229–252 (2005) · Zbl 1086.08002
[5] 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). · Zbl 1151.18001
[6] Jech, T.: Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics (2002).
[7] Pultr, A., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland and Academia Praque, Prague (1980). · Zbl 0418.18004
[8] Trnková V.: Some properties of Set-endofunctors. Commentationes Mathematicae Universitatis Carolinae 10, 323–259 (1969) · Zbl 0183.30401
[9] Trnková V.: On descriptive classification of Set-functors I. Commentationes Mathematicae Universitatis Carolinae 12, 143–174 (1971) · Zbl 0232.18004
[10] Vopěnka P., Pultr A., Hedrlín Z.: A rigid relation exists on any set. Commentationes Mathematicae Universitatis Carlinae 6, 149–155 (1965)
[11] Zmrzlina, A.: Too Many Functors - A continuation of ”The Emergence of Functors”. In: Categorical Perpectives. Trends in Mathematics, pp. 47–62. Birkhauser Verlag (2001) · Zbl 0985.18005
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