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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