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Register a new userUniforming n-place functions on ds(alpha)
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In this paper the Erdos-Rado theorem is generalized to the class of well founded trees.
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It is a survey of the main results on abstract characterizations of algebras of $n$-place functions obtained in the last 40 years. A special attention is paid to those algebras of $n$-place functions which are strongly connected with groups and semigroups, and to algebras of functions closed with respect natural relations defined on their domains.
Harrington and Soare introduced the notion of an n-tardy set. They showed that there is a nonempty $mathcal{E}$ property Q(A) such that if Q(A) then A is 2-tardy. Since they also showed no 2-tardy set is complete, Harrington and Soare showed that there exists an orbit of computably enumerable sets such that every set in that orbit is incomplete. Our study of n-tardy sets takes off from where Harrington and Soare left off. We answer all the open questions asked by Harrington and Soare about n-tardy sets. We show there is a 3-tardy set A that is not computed by any 2-tardy set B. We also show that there are nonempty $mathcal{E}$ properties $Q_n(A)$ such that if $Q_n(A)$ then A is properly n-tardy.
Algebraic properties of $n$-place opening operations on a fixed set are described. Conditions under which a Menger algebra of rank $n$ can be represented by $n$-place opening operations are found.
When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words a subset of). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some $n$, the class consists of the potentially partial recursive functions of $n$ variables, and the collection consists of sets of $n$-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and the collection consists of subsets of the first of them (the condition is also necessary under a weak assumption on the second one). The third theorem is of a similar character but, instead of continuity, it concerns computability in the spirit of the one in effective topological spaces.
We define a family of three related reducibilities, $leq_T$, $leq_{tt}$ and $leq_m$, for arbitrary functions $f,g:Xrightarrowmathbb R$, where $X$ is a compact separable metric space. The $equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $alpha$-jump functions $j_alpha:2^omegarightarrow mathbb R$ are $leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $leq_{tt}$ and $leq_m$, finding an exact match to the $alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.
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