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

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 ${\mathcal{P}\mathcal{H}}_{\mathrm{cf}}^{+}(\mathbb{R})$ is factorizable and $F$-inverse. We describe the structure of the band of the semigroup ${\mathcal{P}\mathcal{H}}_{\mathrm{cf}}^{+}(\mathbb{R})$, its two-sided ideals, maximal subgroups and Green's relations. We prove that the quotient semigroup ${\mathcal{P}\mathcal{H}}_{\mathrm{cf}}^{+}(\mathbb{R})/{\mathfrak{C}}_{\text{mg}}$, where ${\mathfrak{C}}_{\text{mg}}$ is the maximum group congruence on ${\mathcal{P}\mathcal{H}}_{\mathrm{cf}}^{+}(\mathbb{R})/{\mathfrak{C}}_{\text{mg}}$, is isomorphic to the group of all oriental homeomorphisms of the space $\mathbb{R}$, and showe that the semigroup ${\mathcal{P}\mathcal{H}}_{\mathrm{cf}}^{+}(\mathbb{R})$ is isomorphic to a semidirect product ${\mathcal{H}}^{+}(\mathbb{R}){⋉}_{\mathfrak{h}}{\mathcal{P}}_{\infty }(\mathbb{R})$ of the free semilattice with unit $({\mathcal{P}}_{\infty }(\mathbb{R}),\cup )$ by the group ${\mathcal{H}}^{+}(\mathbb{R})$.

11 pages, in Ukrainian