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orthodox semigroup
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.〔J. Almeida, J.-E. Pin and P. Weil (Semigroups whose idempotents form a subsemigroup ) updated version of 〕 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.
==Examples==
*Consider the binary operation in the set ''S'' = defined by the following Cayley table :
:Then ''S'' is an orthodox semigroup under this operation, the subsemigroup of idempotents being .〔
*Inverse semigroups and bands are examples of orthodox semigroups.
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