Invertible element

of a semi-group with identity

An element $x$ for which there exists an element $y$ such that $xy=1$ (right invertibility) or $yx=1$ (left invertibility). If an element is invertible on both the right and the left, it is called two-sidedly invertible (often simply invertible). The set $G(S)$ of all elements with a two-sided inverse in a semi-group $S$ with identity is the largest subgroup in $S$ that contains the identity. A bicyclic semi-group provides an example of the existence of elements that are invertible only on the right or only on the left; in addition, the existence of such elements in a semi-group $S$ implies the existence in $S$ of a bicyclic sub-semi-group with the same identity as $S$. An alternative situation is that in which every element in $S$ with a one-sided inverse has a two-sided inverse; this holds if and only if either $S=G(S)$ or if $S\setminus G(S)$ is a sub-semi-group (being, clearly, the largest ideal in $S$); such a semi-group is called a semi-group with isolated group part. The following are examples of semi-groups with isolated group part: every finite semi-group with identity, every commutative semi-group with identity, every semi-group with two-sided cancellation and identity, and every multiplicative semi-group of complex matrices containing the identity matrix.

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961)
[2]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)