On oriented alternating inverse monoids

Key Points

• The aim is to explore the properties and structures of oriented inverse monoids and their cardinalities.
• Computed the cardinalities of the submonoids \( \mathcal{AOR}_n \) and \( \mathcal{AOP}_n \)
• Described the Green’s structures associated with these monoids
• Analyzed the congruences related to \( \mathcal{AOR}_n \) and \( \mathcal{AOP}_n \)
• Calculated the ranks of \( \mathcal{AOR}_n \) and \( \mathcal{AOP}_n \)
• The cardinalities for \( \mathcal{AOR}_n \) and \( \mathcal{AOP}_n \) were determined explicitly.
• Green's structures revealed interesting properties of these transformations.
• Congruences displayed specific traits unique to oriented transformations.
• Ranks provided insights into the algebraic complexity of the studied monoids.