# Hilbert modules over function algebras by R. G. Douglas, V.I. Paulsen

By R. G. Douglas, V.I. Paulsen

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Assume that from an index relative to A for G we can effectively obtain an index for the covering class S ⊇ G. To compute A n , let G = 2N − [A n ]. Compute the index for S. Wait for a stage when all strings y of length n except for one satisfy [y] ⊆ S. Then A n must be the remaining string. K-triviality. This property of sets is the opposite of ML-randomness: K-trivial sets are “antirandom”. 4, Z is ML-random iff all values K(Z n ) are near their upper bound n + K(n); on the other hand Z is K-trivial if the values K(Z n ) are at their lower bound K(n) (all within constants).

Pages 209–218. Amer. Math. , Providence, RI, 1984. [67] . Trees and linearly ordered sets. In Handbook of set-theoretic topology, pages 235–293. North-Holland, Amsterdam, 1984. [68] Trans. Amer. Math. , . Directed sets and cofinal types. Trans. Amer. Math. , 209:711–723, 1985. [69] . Partitioning pairs of countable ordinals. , 159(3–4):261–294, 1987. [70] . Partition Problems In Topology. Amer. Math. , 1989. Aronszajn orderings. Publ. Inst. Math. ), 57(71):29–46, 1995. ¯Duro Kurepa memorial volume.

Recall that {0, 1}∗ denotes the set of strings over {0, 1}. A machine is a partial computable function M : {0, 1}∗ → {0, 1}∗ . If M (σ) = x we say that σ is an M -description of x. We say that a machine M is prefix-free if no M -description is a proper initial segment of any other M -description. To build a prefix-free machine M , one usually specifies a set of requests r, y ∈ N × {0, 1}∗ . Via such a request one asks that M can describe the string y with r bits. e. set of requests can be turned into a prefix-free machine.