Generalisations of Inverse Semigroups
As noted above, an inverse semigroup S can be defined by the conditions (1) S is a regular semigroup, and (2) the idempotents in S commute; this has led to two distinct classes of generalisations of an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
Examples of regular generalisations of an inverse semigroup are:
- Regular semigroups: a semigroup S is regular if every element has at least one inverse; equivalently, for each a in S, there is an x in S such that axa = a.
- Locally inverse semigroups: a regular semigroup S is locally inverse if eSe is an inverse semigroup, for each idempotent e.
- Orthodox semigroups: a regular semigroup S is orthodox if its subset of idempotents forms a subsemigroup.
- Generalised inverse semigroups: a regular semigroup S is called a generalised inverse semigroup if its idempotents form a normal band, i.e., xyzx = xzyx, for all idempotents x, y, z.
The class of generalised inverse semigroups is the intersection of the class of locally inverse semigroups and the class of orthodox semigroups.
Amongst the non-regular generalisations of an inverse semigroup are:
- (Left, right, two-sided) adequate semigroups.
- (Left, right, two-sided) ample semigroups.
- (Left, right, two-sided) semiadequate semigroups.
- Weakly (left, right, two-sided) ample semigroups.
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