# example of monoids

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?

The dyadic rationals (the rational numbers whose denominator is a natural power of $$2$$) under multiplication has infinitely many invertible elements (any integral power of $$2$$), yet any dyadic rational with an odd prime factor in its numerator is non-regular.
Minimal example. Let $$M = \{1,a,b,0\}$$ be the monoid in which $$1$$ is the identity, $$0$$ is a zero, $$a^2 = 1$$, $$b^2 = 0$$ and $$ab = ba = b$$. The group of units of $$M$$ is $$\{1, a\}$$, the cyclic group of order $$2$$ and $$b$$ is the unique non-regular element.
To obtain $$M$$ as a transformation monoid, just take its right representation. $$\begin{array}{c|c|c|c|c|} &1&2&3&4\\ \hline a&2&1&3&4\\ \hline b&3&3&4&4\\ \hline \end{array}$$