Proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea used a constructive semigroup with inequality in . This motivated us to achieve a little progress in that direction. The starting point is the structure (S,=,≠,·) called a semigroup with apartness. Our primary objective is to prove isomorphism theorems for such constructive semigroups. In doing so our main ideas and notions come from .

We examine basic notions of special subsets and orders in the context of semigroups with apartness and prove constructive analogues of some classical theorems relating such subsets and orders.

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A semigroup S is called surjective if S 2 = S . The aim of this paper is to prove that pn -sequences of finite surjective semigroups are eventually strictly increasing, except in few well known cases, when they are bounded. Also, some further types of finite semigroups, obtained by means of subdirect products, are shown to have the same property.