A Survey of Clones Closed Under Conjugation

A survey of clones on infinite sets

“…It might be interesting to note that on finite X, all symmetric clones are known ([Kho92], [Kho93], [Kho94], [Mar96b], [Mar96a], see also the survey paper [Sze04]). If X has at least five elements, then the only symmetric precomplete clone is the S lupecki-clone of all functions which are either essentially unary or take at most |X| − 1 values.…”

Clones Closed Under Conjugation I: Clones With Constants

“…We will follow the treatment in [6], Chapter 9-10. 24 . This Steiner system is unique, up to isomorphism, and is determined by the binary Golay code, which can be defined as follows.…”

“…(Hence, in case R = End( K A), End( K A) can be replaced by the ring of d × d matrices over K.) It is easy to check (see e.g. [24], Example 2.11) that whether R = K or R = End( K A), the weak automorphism group of the algebra R A c is the affine semilinear group AΓL( K A) = AΓL(d, q). Since AGL(1, 3) = S 3 and AΓL(1, 4) = AGL(2, 2) = S 4 , the clones listed in (A 1 ) and (A 2 ) are G-closed for the groups G indicated.…”

M-Solid Varieties of Algebras

“…The theory of conjugate pairs of completely additive closure operators was used in [16] to characterize lattices of M-solid quasivarieties and in [81] to characterize lattices of M-solid pseudovarieties. A completely different application of this theory is discussed in [96]. Here the basic Galois connection is the connection (Pol, Inv) between operations and relations defined on the same set A and induced by the relation "an operation f preserves a relation Q".…”