Positive equivalences with finite classes and related algebras (Q1204783)

Let \(\eta\) be an equivalence over \(\mathbb{N}\); \(\eta\) is called effectively infinite if there is an infinite recursively enumerable set of pairwise non \(\eta\)-equivalent numbers. An algebra is effectively infinite if its numerating equivalence is effectively infinite. It is proved that a positive equivalence is effectively infinite and there is a finitely generated algebra over it if the powers of its classes are bounded by some natural number. Some examples of non-effectively infinite positive algebras locally finite or finitely generated with finite classes of the numerating equivalences are given.