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