On subamalgams of partially ordered monoids

The study of pomonoid amalgams was initiated by Fakhuruddin in the 1980s and subsequently extended by Bulman-Fleming, Sohail and the authors in the 2000s. We further investigate pomonoids amalgams and in particular we consider the concept of subpomonoid amalgams possessing a suitable ordered version of the unitary property. If [U;T1,T2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[U; T_1, T_2]$$\end{document} is an amalgam of subpomonoids of the amalgam [U;S1,S2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[U; S_1, S_2]$$\end{document} we consider the question of whether the free product of the pomonoid amalgam [U;T1,T2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[U; T_1, T_2]$$\end{document} is poembeddable in the free product of the pomonoid amalgam [U;S1,S2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[U; S_1, S_2]$$\end{document}, giving a sufficient condition in terms of strongly pounitary subpomonoids.