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In the area of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
known as
semigroup theory In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the ...
, an ''E''-semigroup is a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
in which the
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s form a
subsemigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the ...
. Certain classes of ''E''-semigroups have been studied long before the more general class, in particular, a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
that is also an ''E''-semigroup is known as an orthodox semigroup. Weipoltshammer proved that the notion of
weak inverse In mathematics, the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroups, a weak inverse of an element ''x'' in a semigroup is an element ''y'' such that . If every element has a weak inverse, the se ...
(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''(''a'') ≝ is the set of weak inverses of ''a''.


References

Semigroup theory Algebraic structures {{algebra-stub