Absolute flatness and amalgamation in pomonoids

Absolute flatness and amalgamation for partially ordered monoids (briefly pomonoids) were first considered in the mid 1980s by S.M. Fakhruddin in two research articles. Though the study of absolute flatness for pomonoids was revived by X. Shi, S. Bulman-Fleming and others after a dormancy period of almost two decades—resulting in the appearance of several research articles on the subject since 2005—amalgamation in pomonoids was never reconsidered until the recent past when S. Bulman-Fleming and the author produced two research articles on the subject. The primary objectives of these papers were to show that imposition of order subjects the amalgamation of monoids to severe restrictions and to prove that partially ordered groups (briefly pogroups) are amalgamation bases in the class of all pomonoids. Proceeding further, we establish in this article the amalgamation property for the class of pogroups. (The property was first proved for the class of groups by O. Schreier, Abh. Math. Semin. Univ. Hamb. 5:161–183, 1927.) In addition, we show that absolutely flat commutative pomonoids are (strong) amalgamation bases in the category of commutative pomonoids. (A similar result was proved by Fakhruddin for weak amalgamation.) The special amalgamation property and the existence of pushouts in the category of pomonoids, which have been instrumental in proving our main results, are also established.