View source for Invertible element

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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc.  (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>