The free F-restriction semigroups

Abstract

<jats:p> We provide a geometric model for the free <jats:italic>X</jats:italic> -generated <jats:italic>F</jats:italic> -restriction semigroup in the extended signature <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${(\cdot , ^+,^{\mathfrak {m}},\lambda )}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:msup> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , where the unary operation <jats:sup>m</jats:sup> maps an element <jats:italic>a</jats:italic> to the maximum element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$a^{\mathfrak {m}}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> of its <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\sigma $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -class, and the constant <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\lambda $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$X^*$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> with respect to the extended set of generators <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$X\cup \overline{X^*}$$</jats:tex-math> <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:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , where the generators from <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\overline{X^*}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> </jats:alternatives> </jats:inline-formula> are in a bijection with the free monoid <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$X^*$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> and serve to capture the maximum elements of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\sigma $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -classes of the quotient. We also provide models for the free <jats:italic>X</jats:italic> -generated strong and perfect <jats:italic>F</jats:italic> -restriction semigroups in the same extended signature. The constructed models enable us to solve the word problems for all the free objects under consideration. </jats:p>