A New Approach to Universal F-inverse Monoids in Enriched Signature

Abstract We show that the universal X -generated F -inverse monoid F ( G ), where G is an X -generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion $$M(G, X\cup {\overline{G}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∪</mml:mo> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of G , with respect to the extended generating set $$X\cup {\overline{G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∪</mml:mo> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> , where $${\overline{G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is a bijective copy of G which encodes the m -operation in F ( G ). The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph $${\text {Cay}}(G, X\cup {{\overline{G}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Cay</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∪</mml:mo> <mml:mover> <mml:mi>G</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and leads to a new and simpler proof of the universal property of F ( G ).