On oriented alternating inverse monoids

Abstract In this paper, we consider the inverse submonoids $$\mathcal {AOR}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>AOR</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> of oriented transformations and $$\mathcal {AOP}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>AOP</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> of orientation-preserving transformations of the alternating inverse monoid $$\mathcal{A}\mathcal{I}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> on a chain with n elements. We compute the cardinalities, describe the Green’s structures and the congruences, and calculate the ranks of $$\mathcal {AOR}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>AOR</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> and $$\mathcal {AOP}_n.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>AOP</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>