The free F-restriction semigroups

Abstract

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