Clones closed under conjugation
by
Ágnes Szendrei
Bolyai Institute, University of Szeged, Hungary; and University of Colorado at Boulder, USA

Let G be a permutation group acting on a set A. A clone C of operations on A is called G-closed if for every operation f(x₁, ... , x_n) in C, all conjugates ^gf(x₁, ... , x_n)=gf(g^-1(x₁), ... , g^-1(x_n)) of f by permutations g in G also belong to C. Examples of G-closed clones include the clone of all operations, Slupecki's and Burle's clones as well as all clones of homogeneous operations.

The question we are interested in is the following: For which groups G on a finite set A (|A| > 2) is the lattice of G-closed clones finite? L. Szabó (1999) proved that if G acts transitively on the 2-element subsets of A, then a G-closed clone is either (i) idempotent, or (ii) a subclone of Slupecki's clone that contains all constants, or (iii) one of finitely many unary, affine, or quasiprimal clones. By a recent result of K. Kearnes and the speaker, the number of G-closed clones that contain all constants is finite if and only if G is the symmetric or alternating group on A or G is one of the groups AGL(1, 5) (|A|=5), PSL(2, 5) or PGL(2, 5) (|A|=6), PGL(2, 7) (|A|=8), PGL(2, 8) or P\GammaL(2, 8) (|A|=9).

To answer the question stated above, it remains to determine: For which of these groups G is the number of idempotent G-closed clones finite? This question will be the focus of the talk.

Date received: May 5, 2004