## List of issues > Series «Mathematics». 2017. Vol. 21

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On Some Maximal Partial Ultraclones on a Two-Element Set

Multifunctions on a two-element set are considered in this paper. Functions from finite set to set of all subsets of this set are called multifunctions. It is obvious that the superposition in the usual sense not appropriate for multifunctions, therefore, we need to expand the standard concept of superposition. Sets of multifunction closed with respect to the operation of ”expanded” superposition are called multiclones and partial ultraclones depending on the type of superposition.

In the theory of discrete functions the classical problem is description of lattice of clones. Because of difficulty of this problem lattice fragments are studied, for example, the minimum and maximum elements, different intervals. In particular, we note that the descriptions of all maximal clones are known for *k*-valued logic functions, partial functions on *k*-element sets, the descriptions of all maximal hyperclones and ultraclones on a two-element set, multiclones on a two-element set are known. In this work the problem of description of of some maximal ultraclones on a two-element set is considered.

1. Badmaev S.A., Sharankhaev I.K. On Maximal Clones of Partial Ultrafunctions on a Two-element Set (in Russian). Izvestiya Irk. Gos. Univ. Ser. Matematika, 2016, vol. 16, pp. 3-18.

2. Badmaev S.A. On Complete Sets of Partial Ultrafunctions on a Two-element Set (in Russian). Vestnik Buryat. Gos. Univ. Matem., Inform., 2015, no 3, pp. 61-67.

3. Lo Czu Kai. Maximal closed classes in the set of partial functions on multi valued logic. Kiberneticheskiy Sbornik. Novaya seriya., 1988, no 25, pp. 131-141.

4. Panteleyev V.I. Completeness Criterion for Incompletely Defined Boolean Functions (in Russian). Vestnik Samar. Gos. Univ. Est.-Naush. Ser., 2009, vol. 2, no 68, pp. 60-79.

5. Panteleyev V.I. Completeness Criterion for Sub-defined Partial Boolean Functions (in Russian). Vestnik Novosibir. Gos. Univ. Ser.: Matem., Mechan., Inform., 2009, vol. 9, no 3, pp. 95-114.

6. Panteleyev V.I. On Two Maximal Multiclones and Partial Ultraclones (in Russian). Izvestiya Irk. Gos. Univ. Ser. Matematika, 2012, vol. 5, no 4, pp. 46-53.

7. Tarasov V. V. Completeness Criterion for Partial Logic Functions (in Russian). Problemy Kibernetiki, 1975, vol. 30, pp. 319-325.

8. Khalbashkeeva T. U. Some Maximal Ulatraclones of Range 2 (in Russian). Proceedings of the 50th ISSC Students and Progress in Science and Technology. Math, 2012, p. 20.

9. Khalbashkeeva T. U. On Some Partial Ultraclones (in Russian). Proceedings of the 4th Russian School Workshop Syntax and Semantics of Logical Systems, 2012, pp. 129-131.

10. Haddad L., Rosenberg I. G., Schweigert D. A Maximal Partial Clone and Slupecki-type Criterion. Acta Sci. Math., 1990, vol. 54, pp 89-98.

11. Rosenberg I. G. Uber die Verschiedenheit Maximaler Klassen in Pk. Rev. Roumaine Math. Pures Appl., 1969, vol. 14, pp. 431-438.