A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page

Very many clones above the unary clone

Let c:= 2א0. We give a family of pairwise incomparable clones on ℕ with 2c members, all with the same unary fragment, namely the set of all unary operations. We also give, for each n, a family of 2c clones all with the same n-ary fragment, and all containing the set of all unary operations. © 2013 Springer Basel