Lower Semilattices of Separable Congruences of Numbered Algebras

Under study are the closure properties for various classes of separable congruences (in particular, of equivalences) of numbered algebras on the natural numbers under the upper and lower bounds in the lattice of congruences. We show that the class of all positive congruences forms a sublattice whereas the classes of negative, computably separable, and separable congruences form a lower but not always an upper subsemilattice. It is shown that uniformly computably separable congruences can fail to form lower and upper semilattices. We give a characterization of semienumerable sets in terms of uniformly computably separable equivalences.

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