# Thread: inverse semigroup

1. ## inverse semigroup

Hi everyone,
My name is Bantom, and I need help for my thesis which is about inverse semigroup. Would anyone here by any chance happen to know :
1. I need some examples of inverse semigroup that are not groups ?
2. And also, some examples of free inverse semigroups?

I would appreciate any input upon this matter. Thanks in advance!

2. Originally Posted by bantom
Hi everyone,
My name is Bantom, and I need help for my thesis which is about inverse semigroup. Would anyone here by any chance happen to know :
1. I need some examples of inverse semigroup that are not groups ?
2. And also, some examples of free inverse semigroups?

I would appreciate any input upon this matter. Thanks in advance!
The Bicyclic Monoid! Also, semilattices are inverse (they are regular because of the idempotents (every band is regular) and as they are commutative all their idempotents commute. Thus, they are inverse).

Bicyclic Monoid: The bicyclic monoid is the monoid given by the presentation $\langle b, c; bc=1 \rangle$.

Semilattice: A semilattice is a commutative semigroup of idempotents. For example, take a set, $S$. Then the set of all finite subsets of $S$ forms a semigroup under the operation of union. This operation is commutative, and all elements are idempotents. I believe that this is the free join-semilattice. Note that there is a formal definition using the meet' operation.

Howie's rather fine book, Fundamentals of Semigroup Theory', contains a chapter on inverse semigroups, and Mark Lawson has a book in this area (it's called something like inverse semigroups: the theory of partial symmetry'). If you are doing a thesis about semigroups then I presume someone in your department studies them, so Howie's book will be in your uni's library. Lawson's may or may not be (apparently it is in mine!)

3. can you tell me some specific examples of inverse semigroup, if possible a popular one, and please describe the examples in these manner (S,*,-1) which is the set and their elements, their operation, and their rule of inverse. I'm sorry for requesting a lot of specifics, but due to my limited knowledge it would be very helpful if the examples are presented in that manner. Thanks in advance!

4. Originally Posted by bantom
can you tell me some specific examples of inverse semigroup, if possible a popular one, and please describe the examples in these manner (S,*,-1) which is the set and their elements, their operation, and their rule of inverse. I'm sorry for requesting a lot of specifics, but due to my limited knowledge it would be very helpful if the examples are presented in that manner. Thanks in advance!
Let $X$ be a set. Now, define $S=\{\gamma; \gamma \subseteq X, |\gamma| < \infty \}$. That is, $S$ consists of all finite subsets of $X$.

Then taking $S$ under the operation of union gives you an inverse semigroup. I will leave it to you to prove why.

Now, I really would recommend reading the excellent book by John Howie. It has an entire section on inverse semigroups, and is an all-round fantastic book on the subject of semigroups!

EDIT: Reading your second post there, I am a tad confused. What do you mean by rule of inverse'? There is no rule of inverse - a semigroup is inverse if for all $x$ there exists a unique $y$ such that $xyx=x$ and $yxy=y$.