Proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea used a constructive semigroup with inequality in [8]. 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 [10].
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