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Partial clones containing all Boolean monotone self-dual partial functions. (English) Zbl 1394.08002

Summary: The study of partial clones on \(\mathbf{2}:= \{0, 1\}\) was initiated by R. V. Freivald. In his fundamental paper published in [Sov. Phys., Dokl. 11, 288–289 (1966; Zbl 0149.24405); translation from Dokl. Akad. Nauk SSSR 167, 1249–1250 (1966)], R. V. Frejvald showed, among other things, that the set of all monotone partial functions and the set of all self-dual partial functions are both maximal partial clones on \(\mathbf{2}\).
Several papers dealing with intersections of maximal partial clones on 2 have appeared after Freivald work. It is known that there are infinitely many partial clones that contain the set of all monotone selfdual partial functions on \(\mathbf{2}\), and the problem of describing them all was posed by some authors.
In this paper we show that the set of partial clones that contain all monotone self-dual partial functions on \(\mathbf{2}\) is of continuum cardinality.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
08A55 Partial algebras

Citations:

Zbl 0149.24405
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