The quasi-uniform character of a topological semigroup

The topological embedding of a topological semigroup S  , commutative with the property of cancelation, into the group G=S×S/RG=S×S/R, (R   the equivalence (a,b)R(a′,b′)⇔ab′=a′b(a,b)R(a′,b′)⇔ab′=a′b) to which S is algebraically embedded, was the subject of the search for the mathematicians of a long period. It was based on the fact that S must naturally be a uniform topological space, as every topological group was. The present paper is devoted to the fact that a quasi-uniformity is defined to any topological space, thus to any topological semigroup.