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E-semigroup
In the area of mathematics known as semigroup theory, an ''E''-semigroup is a semigroup in which the idempotents form a subsemigroup.
Certain classes of ''E''-semigroups have been studied long before the more general class, in particular, a regular semigroup that is also an ''E''-semigroup is known as an orthodox semigroup.
Weipoltshammer proved that the notion of weak inverse (the existence of which is one way to define ''E''-inversive semigroups) can also be used to define/characterize ''E''-semigroups as follows: a semigroup ''S'' is an ''E''-semigroup if and only if, for all ''a'' and ''b'' ∈ ''S'', ''W''(''ab'') = ''W''(''b'')''W''(''a''), where ''W''(''x'') ≝ is the set of weak inverses of ''x''.〔
== References ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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