Uncountable families of partial clones containing maximal clones
The authors prove that for every maximal clone on a finite set determined by a prime affine or an \(h\)-universal relation, the number of partial clones containing this clone equals the size of the continuum. A partial clone on a set \(X\) is a composition closed set of partial finitary operations on \(X\) which contains the projections. A total clone is a partial clone which contains only total operations. The lattice of partial clones can be partitioned as follows: To every total clone \(C\), assign the set \(P_C\) of those partial clones whose total operations are precisely \(C\). The problem of determining such equivalence classes was initiated by \textit{V. B. Alekseev} and \textit{A. A. Voronenko} [Discrete Math. Appl. 4, No. 5, 401--419 (1994); translation from Diskretn. Mat. 6, No. 4, 58--79 (1994; Zbl 0818.06013)]. In a paper by \textit{L. Haddad}, \textit{D. Lau} and \textit{I. G. Rosenberg} yet to be published, the problem was considered for maximal clones \(C\), that is, for clones \(C\) which are dual atoms in the lattice of (total) clones. All maximal clones were determined by \textit{I. Rosenberg} [Rozpr. Česk. Akad. Věd, Řada Mat. Přír. Věd 80, No. 4, 94 p. (1970; Zbl 0199.30201)], and were divided into six classes. For clones \(C\) in four of those classes, the problem of determining the structure of \(P_C\) was partially settled in the above-mentioned yet unpublished paper. In the present paper, the authors show that \(P_C\) is of size continuum for all \(C\) in the remaining two classes: These comprise the polymorphism clones of prime affine relations and of \(h\)-universal relations.
scientific article; zbMATH DE number 5152754