Editing Syntactic monoid (section)

==Syntactic quotient==
An [[Alphabet (formal languages)|'''alphabet''']] is a finite [[Set (mathematics)|set]].

The [[free monoid|'''free monoid''']] on a given alphabet is the monoid whose elements are all the [[string (computer science)|strings]] of zero or more elements from that set, with [[string concatenation]] as the monoid operation and the [[empty string]] as the [[identity element]].

Given a [[subset]] <math>S</math> of a free monoid <math>M</math>, one may define sets that consist of formal left or right [[Inverse element#In a unital magma|'''inverses''' of elements]] in <math>S</math>.  These are called [[Quotient of a formal language|quotients]], and one may define right or left quotients, depending on which side one is concatenating. Thus, the '''right quotient''' of <math>S</math> by an element <math>m</math> from <math>M</math> is the set
:<math>S \ / \ m=\{u\in M \;\vert\; um\in S \}.</math>

Similarly, the '''left quotient''' is

:<math>m \setminus S=\{u\in M \;\vert\; mu\in S \}.</math>