  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 $\mathscr{H}^+\!(\mathbb{R})$.
