Definition:Inverse Subset

From ProofWiki

Jump to: navigation, search

Definition

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

Let $C \subseteq S$ be the set of cancellable elements of $S$.

Let $X \subseteq C$.

Then the inverse of the subset $X$ is defined as:

$X^{-1} = \left\{{y \in S: x \in X, x \circ y = e_S}\right\}$

That is, it is the set of all the inverses of all the elements in the set $X$.

When $\left({G, \circ}\right)$ is a group, then $C = G$ as from the Cancellation Laws, all group elements are cancallable.

Thus $X^{-1}$ is then defined as:

$X^{-1} = \left\{{x^{-1} \in G: x \in X}\right\}$