It is a theorem that every commutative monoid is inversible, i.e. is isomorphic with a submonoid of a(commutative) group. It is also clear that a group contains all submonoids generated by any subset of its underlying set. but it is also known that non every monoid is inversible, i.e. cannot be isomorphically imbedded inside some group. Question:Are there simple examples of noninversible monoids ? Gérard Lang

A complete description of monoids embeddable into groups was given by Malcev more than 70 years ago. The description is in a form of infinitely many quasiidentities, the easiest of those are the two cancelative laws. See Chapter 12 of Clifford, A. H.; Preston, G. B. The algebraic theory of semigroups. Vol. II. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I. 1967 xv+350 pp. 


An example is given here: Counterexamples in Algebra? To write it out here, take the monoid $\langle a,b,c,d,x,y,u,v : ax=by, cx=dy, au=bv \rangle$. 


Let $M$ be a totally ordered set of cardinal $\ge3$ with a least element $\bot$, and we define a binary operation $x\cdot{}y=\max(x,y)$. This is obviously a commutative monoid with $\bot$ as the identity element, but it is not invertible because if $c\lt{}b\lt{}a$ you have $a\cdot{}b=a\cdot{}c$ but $b\neq{}c$. 

