The First Isomorphism Theorem for (Co-ordered) $\Gamma$-Semigroups with Apartness

The First-Order Isomorphism Theorem

For any class C und closed under NC reductions, it is shown that all sets complete for C under first-order (equivalently, Dlogtimeuniform AC) reductions are isomorphic under first-order computable

An Introduction to Implicative Semigroups with Apartness

The setting of this research is Bishop’s constructive mathematics. Following ideas of Chan and Shum, exposed in their famous paper “Homomorphisms of implicative semigroups”, we discuss the structure of implicative semigroups on sets with tight apartness. Moreover, we use anti-orders instead of partial orders. We study concomitant issues induced by existence of apartness and anti-orders giving s...

A Theorem on Anti - Ordered Factor - Semigroups

Let K be an anti-ideal of a semigroup (S,=, =, ·, θ) with apartness. A construction of the anti-congruence Q(K) and the quasi-antiorder θ, generated by K, are presented. Besides, a construction of the anti-order relation Θ on syntactic semigroup S/Q(K), generated by θ, is given in Bishop’s constructive mathematics.

A First-Order Isomorphism Theorem

We show that for most complexity classes of interest, all sets complete under rstorder projections (fops) are isomorphic under rst-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds. Since \natural" complete problems seem to stay complete via fops, this indicates that up to rst-order isomorphism there is only one \natural" complete problem for each \...

Ordered Categories and Ordered Semigroups