F-birestriction monoids in enriched signature

Abstract Motivated by recent interest to F -inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of F -birestriction monoids as algebraic structures in the enriched signature $$(\cdot , \, ^*, \,^+, \,\phantom {0}^{\mathfrak {m}},1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mmultiscripts> <mml:mspace/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> <mml:mo>,</mml:mo> <mml:mmultiscripts> <mml:mspace/> <mml:mrow/> <mml:mo>+</mml:mo> </mml:mmultiscripts> <mml:mo>,</mml:mo> <mml:mspace/> <mml:msup> <mml:mphantom> <mml:mn>0</mml:mn> </mml:mphantom> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where the unary operation $$(\cdot )^{\mathfrak {m}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:math> maps each element to the maximum element of its $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -class. We find a presentation of the free F -birestriction monoid $${\textsf{FFBR}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>FFBR</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> as a birestriction monoid $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> over 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 $$\overline{X^+}$$ <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> is a set in a bijection with the free semigroup $$X^+$$ <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 encodes the maximum elements of (non-projection) $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -classes. This enables us to show that $${\textsf{FFBR}}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>FFBR</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> decomposes as the partial action product $$E(\mathcal {I})\rtimes X^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⋊</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> of the idempotent semilattice of the universal inverse monoid $$\mathcal {I}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> of $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> partially acted upon by the free monoid <jats:tex-m