5 research outputs found
Boolean algebras realized by c.e. equivalence relations
В статье рассматривается связь между алгоритмическими свойствами отношений эквивалентности и булевыми алгебрами, реализованными этими отношениями эквивалентности
Joins and meets in the structure of Ceers
We study computably enumerable equivalence relations (abbreviated as ceers)
under computable reducibility, and we investigate the resulting degree
structure Ceers, which is a poset with a smallest and a greatest element. We
point out a partition of the ceers into three classes: the finite ceers, the
light ceers, and the dark ceers. These classes yield a partition of the degree
structure as well, and in the language of posets the corresponding classes of
degrees are first order definable within Ceers. There is no least, no maximal,
no greatest dark degree, but there are infinitely many minimal dark degrees. We
study joins and meets in Ceers, addressing the cases when two incomparable
degrees of ceers X,Y have or do not have join or meet according to where X,Y
are located in the classes of the aforementioned partition: in particular no
pair of dark ceers has join, and no pair in which at least one ceer is dark has
meet. We also exhibit examples of ceers X,Y having join which coincides with
their uniform join, but also examples when their join is strictly less than the
uniform join. We study join-irreducibility and meet-irreducibility. In
particular we characterize the property of being meet-irreducible for a ceer E,
by showing that it coincides with the property of E being self-full, i.e. every
reducibility from E to itself is in fact surjective on its equivalence classes
(this property properly extends darkness). We then study the quotient structure
obtained by dividing the poset Ceers by the degrees of the finite ceers, and
study joins and meets in this quotient structure. We look at automorphisms of
Ceers, and show that there are continuum many automorphisms fixing the dark
ceers, and continuum many automorphisms fixing the light ceers. Finally, we
compute the complexity of the index sets of the classes of ceers studied in the
paper
Weakly precomplete computably enumerable equivalence relations
Using computable reducibility on equivalence relations, we investigate weakly precomplete ceers (a ``ceer'' is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers , that are not precomplete; and there are infinitely many isomorphism types of non-universal weakly precomplete ceers. Whereas the Visser space of a precomplete ceer always contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete
ceers is -complete, whereas the index set of the weakly precomplete ceers is -complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the -complete ceers are -complete