Abstract
The following result has been shown recently in the form of a dichotomy: For every total clone C on 
:= {0, 1}, the set \({\mathcal{I}}\)(C) of all partial clones on whose total component is C is either finite or of continuum cardinality. In this paper, we show that the dichotomy holds, even if only strong partial clones are considered, i.e., partial clones which are closed under taking subfunctions: For every total clone C on , the set \({\mathcal{I}_{\rm Str}}\)(C) of all strong partial clones on whose total component is C, is either finite or of continuum cardinality.
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References
6, 58–79 (1994). English translation: Alekseev, V.B., Voronenko, L.L.: On some closed classes in partial two-valued logic. Discrete Math. Appl. 4, 401–419 (1994)
16 (1970)Couceiro, M., Haddad, L., Schölzel, K., Waldhauser, T.: A Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones. IEEE 43rd International Symposium on Multiple-Valued Logic (ISMVL), pp. 123–128 (2013)
Creignou, N., Kolaitis, P., Zanuttini, B.: Structure identification of boolean relations and plain bases for co-clones. J. Comput. Syst. Sci. 74, 1103–1115 (2008)
Fugere, J., Haddad, L.: On partial clones containing all idempotent partial operations. 28th IEEE International Symposium on Multiple-Valued Logic, pp. 369–373 (1998)
Haddad, L., Lau, D.: Uncountable families of partial clones containing maximal clones. Beitr. Algebra Geom. 48, 257–280 (2007)
Haddad L., Lau D., Rosenberg I.G.: Intervals of partial clones containing maximal clones. J. Autom. Lang. Comb. 11, 399–421 (2006)
Haddad, L., Simons, G.E.: On intervals of partial clones of Boolean partial functions. 33rd IEEE International Symposium on Multiple-Valued Logic, pp. 315–320 (2003)
Harnau W.: Ein verallgemeinerter Relationenbegriff für die Algebra der mehrwertigen Logik, Teil I (Grundlagen). Rostock. Math. Kolloq. 28, 5–17 (1985)
Harnau, W.: Ein verallgemeinerter Relationenbegriff für die Algebra der mehrwertigen Logik, Teil II (Relationenpaare). Rostock. Math. Kolloq. 31, 11–20 (1987)
Harnau, W.: Ein verallgemeinerter Relationenbegriff für die Algebra der mehrwertigen Logik, Teil III (Beweis). Rostock. Math. Kolloq. 32, 15–24 (1987)
Lau, D.: Function Algebras on Finite Sets. A Basic Course on Many-valued Logic and Clone Theory. Springer Monographs in Mathematics. Springer, Berlin (2006)
Lau, D., Schölzel, K.: A Classification of Partial Boolean Clones. 40th IEEE International Symposium on Multiple-Valued Logic, pp. 189–194 (2010)
2, 1–11 (1981). English translation: Romov, B.A.: The algebras of partial functions and their invariants. Cybernetics, 17, 157–167 (1981)Strauch, B.: Die Menge \({\mathcal{M}(M \cap T_{0} \cap T_{1})}\). Universität Rostock (1995, preprint)
Strauch, B.: Die Menge \({\mathcal{M}(S \cap T_{0} \cap T_{1})}\). Universität Rostock (1996, preprint)
Strauch, B.: Noncountable many classes containing a fixed class of total Boolean functions. In: Denecke, K. (ed.) Proceedings of the conference on general algebra and discrete mathematics, Potsdam, June 1996. Berichte aus der Mathematik, pp. 177–188. Shaker, Aachen (1997)
Strauch, B.: On partial classes containing all monotone and zero-preserving total Boolean functions. Math. Log. Q. 43, 510–524 (1997)
6, 58–79 (1994). English translation: Alekseev, V.B., Voronenko, L.L.: On some closed classes in partial two-valued logic. Discrete Math. Appl. 4, 401–419 (1994)
16 (1970)
2, 1–11 (1981). English translation: Romov, B.A.: The algebras of partial functions and their invariants. Cybernetics, 17, 157–167 (1981)Author information
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Presented by R. Poeschel.
The research of the author was supported by the internal research project MRDO2.
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Schölzel, K. Dichotomy on intervals of strong partial Boolean clones. Algebra Univers. 73, 347–368 (2015). https://doi.org/10.1007/s00012-015-0330-7
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DOI: https://doi.org/10.1007/s00012-015-0330-7