Regular semigroups weakly generated by set

Abstract We prove the existence and uniqueness (up to isomorphism) of a regular semigroup $${{\,\textrm{F}\,}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>F</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> weakly generated by a set X of elements such that all other regular semigroups weakly generated by X are homomorphic images of $${{\,\textrm{F}\,}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>F</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The semigroup $${{\,\textrm{F}\,}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>F</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is introduced by a presentation and the word problem for that presentation is solved. The structure of the semigroup $${{\,\textrm{F}\,}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>F</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is also studied.