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Let be a semigroup.
1 Regular semigroups
is a regular semigroup if all its elements are regular. The phrase ’von Neumann regular’ is sometimes used, after the definition for rings.
In a regular semigroup, every principal ideal is generated by an idempotent.
Every regular element has at least one inverse. To show this, suppose is regular, so that for some . Put . Then
so is an inverse of .
2 Inverse semigroups
is an inverse semigroup if for all there is a unique such that and .
In an inverse semigroup every principal ideal is generated by a unique idempotent.
In an inverse semigroup the set of idempotents is a subsemigroup, in particular a commutative band (http://planetmath.org/ASemilatticeIsACommutativeBand).
The bicyclic semigroup is an example of an inverse semigroup. The symmetric inverse semigroup (on some set ) is another example. Of course, every group is also an inverse semigroup.
Both of these notions generalise the definition of a group. In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents. Apart from inverse semigroups, there are orthodox semigroups where the set of idempotents is a subsemigroup, and Clifford semigroups where the idempotents are central.
is called eventually regular (or -regular) if a power of every element is regular.
is called group-bound (or strongly -regular, or an epigroup) if a power of every element is in a subgroup of .
is called completely regular if every element is in a subgroup of .
|Date of creation||2013-03-22 14:23:17|
|Last modified on||2013-03-22 14:23:17|
|Last modified by||yark (2760)|