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

Clones on infinite sets

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.Comment: 70 pages; Dissertation written at the Vienna University of Technology under the supervision of Martin Goldstern; essentially consists of the author's papers "The clone generated by the median functions", "Clones containing all almost unary functions, "Maximal clones on uncountable sets that include all permutations" which are all available from arXi

Bergman Spaces on Disconnected Domains

For a bounded region G ⊂ ℂ and a compact set K ⊂G , with area measure zero, we will characterize the invariant subspaces M (under ƒ → z ƒ) of the Bergman space Lpa(G\K), 1 ≤ p \u3c ∞, which contain L\u3csup\u3epa(G) and with dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K. When G\K is connected, we will see that dim(M/(z-⋋)M) = 1 for all ⋋ ∈ G\K and this in this case we will have a complete description of the invariant subspaces lying between L\u3csup\u3epa(G) and Lpa(G\K). When G\K is not connected, we will show that in general the invariant subspaces between Lpa(G) and Lpa(G\K) are fantastically complicated. As an application of these results, we will remark on the complexity on the invariant subspaces (under ƒ → ζ ƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space B12(∏), there are invariant subspaces F such that the dimension of ζF in F is infinite