I am reading up on Wheel Theory using the notes found at https://www2.math.su.se/reports/2001/11/2001-11.pdf. There really doesn't seem to be many online notes for this topic. It starts by motivating wheels by looking at the ring of fractions, but I am confused by what it states on page ${4}$:
${A\times S}$ is a product of monoids, with ${\sim_S}$ a congruence relation on it. Let ${\equiv_S}$ be the congruence relation generated on the monoid ${A\times A}$ by ${\sim_S}$. Then ${(x,y)\equiv_S (x',y')\Leftrightarrow \exists\ s,s' \in S: (sx,sy) = (s'x,s'y)}$
What exactly does it mean by ${\equiv_S}$ is generated by ${\sim_S}$?