Semirigid sets of quasilinear clones
Akihiro Nozaki, Grant R. Pogosyan, Masahiro Miyakawa, Ivo G. Rosenberg · 2002
Let k be a prime and G a Galois field on k:=(0,1,. . .,k-1). The set of all quasilinear (or affine with respect to G) k-valued logic functions is a maximal clone called quasilinear. A family of quasilinear clones on k is semirigid if the clones of the family share exactly the constant functions and the projections. Semirigid sets of quasilinear clones are needed for the classification of bases of k-valued logic, which is unknown for k>3. The authors characterize all semirigid sets of quasilinear clones. In particular, for k=5 they describe all semirigid triples of quasilinear clones and show that no such pair exists. For every prime k>5 they exhibit a semirigid pair of quasi-linear clones. The techniques used are based on elementary number theory and on polynomials over G.>