14 research outputs found
Free-algebra functors from a coalgebraic perspective
Given a set of equations, the free-algebra functor
associates to each set of variables the free algebra over
. Extending the notion of \emph{derivative} for an arbitrary set
of equations, originally defined by Dent, Kearnes, and Szendrei, we
show that preserves preimages if and only if , i.e. derives its derivative . If weakly
preserves kernel pairs, then every equation gives rise to a
term such that and . In
this case n-permutable varieties must already be permutable, i.e. Mal'cev.
Conversely, if defines a Mal'cev variety, then weakly
preserves kernel pairs. As a tool, we prove that arbitrary endofunctors
weakly preserve kernel pairs if and only if they weakly preserve pullbacks
of epis
Completeness and Universality of Arithmetical Numberings
Abstract We investigate completeness and universality notions, relative to differ-ent oracles, and the interconnection between these notions, with applica-tions to arithmetical numberings. We prove that principal numberings are complete; completeness is independent of the oracle; the degree of any incomplete numbering is meet-reducible, uniformly complete num-berings exist. We completely characterize which finite arithmetical fam-ilies have a universal numbering
A survey on universal computably enumerable equivalence relations
We review the literature on universal computably enumerable equivalence relations, i.e. the computably enumerable equivalence relations (ceers) which are -complete with respect to computable reducibility on equivalence relations. Special attention will be given to the so-called uniformly effectively inseparable (u.e.i.) ceers, i.e. the nontrivial ceers yielding partitions of the natural numbers in which each pair of distinct equivalence classes is effectively inseparable (uniformly in their representatives). The u.e.i. ceers comprise infinitely many isomorphism types. The relation of provable equivalence in Peano Arithmetic plays an important role in the study and classification of the u.e.i. ceers