Ore Set

A Submonoid $S\subseteq M$ of a Monoid $M$ is a Left Ore Set if it satisfies the following two additional axioms:

• Left cancellability: If $ms=ns$ for $m,n:M$, $s\in S$, then there exists some $s^{\prime}\in S$ such that $s^{\prime}m=s^{\prime}n$.
• Left Ore condition: for any $m:M$ and $s\in S$, there exists some $m^{\prime}:M$ and $s^{\prime}\in S$ such that $s^{\prime}m=m^{\prime}s$.

Dually, $S$ is a Right Ore Set if it satisfies the following dual axioms:

• Right cancellability: If $sm=sn$ for some $m,n:M$, $s\in S$, then there exists some $s^{\prime}\in S$ such that $ms^{\prime}=ns^{\prime}$.
• Right Ore condition: for any $m:M$ and $s\in S$, there exists some $m^{\prime}:M$ and $s^{\prime}\in S$ such that $ms^{\prime}=sm^{\prime}$.

The use of Ore sets is in construction of Localizations of Monoids; Ore sets let us avoid adding lots of formal inverses, and instead simply work with fractions.

Decategorification

Left and right Ore sets are a decategorification of right and left Calculi of Fractions.