An element $x$ of a semigroup $S$ is called regular provided that there exists $y\in S$ such that $xyx=x$. $S$ is called regular if all its elements are regular. Let $S$ be a monoid with identity element $1$. An element $a\in S$ is called invertible if there exists $b\in S$ such that $ab=ba=1$. The set of all inverse elements of the monoid $S$ is denoted by $S^{\star}$.
Are there examples of monoids that are non-regular and have more than one invertible element, particularly related to transformation semigroups?