92 research outputs found
Rogers semilattices of families of two embedded sets in the Ershov hierarchy
Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a su cient condition on a, so that for every 1 a {computable family of two embedded sets, i.e. two sets A;B, with A properly contined in B, the Rogers semilattice of the family is in nite. This condition is satis ed by every notation of !; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satis es this condition. On the other hand, we also give a su cient condition on a,
that yields that there is a 1 a {computable family of two embedded sets,
whose Rogers semilattice consists of exactly one element; this condition is
satis ed by all notations of every successor ordinal bigger than 1, and by
all notations of the ordinal !+!; moreover every computable ordinal that
is sum of two smaller ordinals has a notation that satis es this condition.
We also show that for every nonzero n 2 !, or n = !, and every notation
of a nonzero ordinal there exists a 1 a {computable family of cardinality
n, whose Rogers semilattice consists of exactly one element
Rogers semilattices of families of two embedded sets in the Ershov hierarchy
Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a su cient condition on a, so that for every 1 a {computable family of two embedded sets, i.e. two sets A;B, with A properly contined in B, the Rogers semilattice of the family is in nite. This condition is satis ed by every notation of !; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satis es this condition. On the other hand, we also give a su cient condition on a,
that yields that there is a 1 a {computable family of two embedded sets,
whose Rogers semilattice consists of exactly one element; this condition is
satis ed by all notations of every successor ordinal bigger than 1, and by
all notations of the ordinal !+!; moreover every computable ordinal that
is sum of two smaller ordinals has a notation that satis es this condition.
We also show that for every nonzero n 2 !, or n = !, and every notation
of a nonzero ordinal there exists a 1 a {computable family of cardinality
n, whose Rogers semilattice consists of exactly one element
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Categorically closed topological groups
Let be a subcategory of the category of topologized semigroups
and their partial continuous homomorphisms. An object of the category
is called -closed if for each morphism
of the category the image is closed in . In the paper
we detect topological groups which are -closed for the categories
whose objects are Hausdorff topological (semi)groups and whose
morphisms are isomorphic topological embeddings, injective continuous
homomorphisms, continuous homomorphisms, or partial continuous homomorphisms
with closed domain.Comment: 26 page
Internal Parameterization of Hyperconnected Quotients
One of the most fundamental facts in topos theory is the internal
parameterization of subtoposes: the bijective correspondence between subtoposes
and Lawvere-Tierney topologies. In this paper, we introduce a new but
elementary concept, "a local state classifier," and give an analogous internal
parameterization of hyperconnected quotients (i.e., hyperconnected geometric
morphisms from a topos). As a corollary, we obtain a solution to the Boolean
case of the first problem of Lawvere's open problems.Comment: 44page
Operator on types of numberings
The reductions constructed permit to give various alternative proofs to known results
about d.r.e. and, in general,a-r.e. numberings. For example, one can show that for every recursive ordinal
a which is not an w-power and every Rogers semilattice of r.e. sets, the corresponding semilattice is also
realized within the a-r.e. numberings
Operator on types of numberings
The reductions constructed permit to give various alternative proofs to known results
about d.r.e. and, in general,a-r.e. numberings. For example, one can show that for every recursive ordinal
a which is not an w-power and every Rogers semilattice of r.e. sets, the corresponding semilattice is also
realized within the a-r.e. numberings
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