Regular semigroups weakly generated by set

We prove the existence and uniqueness (up to isomorphism) of a regular semigroup \({{\,\textrm{F}\,}}(X)\) $\text{F}(X)$ weakly generated by a set X of elements such that all other regular semigroups weakly generated by X are homomorphic images of \({{\,\textrm{F}\,}}(X)\) $\text{F}(X)$ . The semigroup \({{\,\textrm{F}\,}}(X)\) $\text{F}(X)$ is introduced by a presentation and the word problem for that presentation is solved. The structure of the semigroup \({{\,\textrm{F}\,}}(X)\) $\text{F}(X)$ is also studied.