# The semigroup of monotone co-finite partial homeomorphisms of the real line

## TL;DR

This paper studies the algebraic structure of the semigroup of monotone co-finite partial homeomorphisms of the real line, revealing its factorizability, structure, and relation to group actions, with implications for understanding its internal symmetries.

## Contribution

It characterizes the semigroup's structure, including its band, ideals, and maximal subgroups, and establishes its isomorphism to a semidirect product involving the group of orientation-preserving homeomorphisms.

## Key findings

- The semigroup is factorizable and $F$-inverse.
- The structure of the band and ideals is described.
- The semigroup is isomorphic to a semidirect product of a free semilattice and a group.

## Abstract

In the paper we investigate the semigroup of monotone co-finite partial homeomorphisms of the space of the usual real line $\mathbb{R}$. We prove that the inverse semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$ is factorizable and $F$-inverse. We describe the structure of the band of the semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$, its two-sided ideals, maximal subgroups and Green's relations. We prove that the quotient semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the maximum group congruence on $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})/\mathfrak{C}_{\textsf{mg}}$, is isomorphic to the group of all oriental homeomorphisms of the space $\mathbb{R}$, and showe that the semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$ is isomorphic to a semidirect product $\mathscr{H}^+\!(\mathbb{R})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{R})$ of the free semilattice with unit $(\mathscr{P}_{\!\infty}(\mathbb{R}),\cup)$ by the group $\mathscr{H}^+\!(\mathbb{R})$.