what is a symmetric semigroup?

Question

First, I think I know what is a symmetric group roughly from algebra. The group of permutation on a set with $n$ element is denoted by $S_n$, and called the symmetric group on $n$ elements (or $n$ numbers). A permutation group is a group $G$ whose elements are permutation of a given set and whose group operation is the composition of permutation in $G$ (which are though of as bijective function from the set $M$ to itself).The group of all permutation of $M$ is the symmetric group of $M$, denoted as Sym$(M)$. (So the term permutation group means a subgroup of symmetric group. Take $M=\{1,2,\cdots,n\}$, the symmetric group on $n$ letters is can be denoted by $S_n$. I also noted there has been several other discussion, so the picture is clear to me.

Now consider the semigroup $S(t)$ on a Hilbert space, which is a family of bounded linear operator satisfies $$ S(t+s)=S(t)\circ S(s),\quad S(0)=I$$

How can I give a definition of a symmetric semigroup ?

Appreciate for any helps.

Answer 1

The semigroup analog of the symmetric group $S(E)$ is the transformation monoid $T(E)$. A transformations on $E$ is a function from $E$ to itself and $T(E)$ is the monoid of all transformations on $E$ under composition.

Edit. Another monoid of interest is the symmetric inverse monoid $I(E)$ consisting of all partial bijections (i.e. partial one-one transformations) on $E$. The following representation theorems hold:

1. Every group can be embedded in a symmetric group.
2. Every inverse monoid can be embedded in a symmetric inverse monoid.
3. Every monoid can be embedded in a transformation monoid.