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In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup. The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox semigroups have been studied earlier. For example, semigroups which are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.
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