On homomorphisms of bicyclic extensions of archimedean totally ordered groups

Let ${\mathcal{B}}^{+}(G)$ be the bicyclic extension of a totally ordered group $G$ which is defined in [O. Gutik, D. Pagon, and K. Pavlyk, Congruences on bicyclic extensions of a linearly ordered group, Acta Comment. Univ. Tartu. Math. 15 (2011), no. 2, 61-80, (arXiv:1111.2401)]. We show that if $G$ and $H$ are archimedean totally ordered groups then every $o$-homomorphism $\widehat{\phi }\text{\hspace{0.17em}⁣}:G\to H$ generates a monoid homomorphim $\tilde{\phi }\text{\hspace{0.17em}⁣}:{\mathcal{B}}^{+}(G)\to {\mathcal{B}}^{+}(H)$, and every monoid homomorphism $\tilde{\phi }\text{\hspace{0.17em}⁣}:{\mathcal{B}}^{+}(G)\to {\mathcal{B}}^{+}(H)$ generates $o$-homomorphism $\widehat{\phi }\text{\hspace{0.17em}⁣}:G\to H$.