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 $\mathscr{ H}^+\!(\mathbb{ R})$.
Categories: math.GR