Unitary posets and amalgamations of pomonoids

Abstract

In 1927, Schreier proved that amalgams of groups are always embeddable in the category of groups. However, this is not true in the category of semigroups, as shown by Kimura. Subsequently, Howie initiated the study of semigroup amalgams by investigating when the embeddablity happens, and found that semigroup amalgams can be embeddable if the core of the amalgam is almost unitary [18]. Later, Hall proved that inverse semigroups are amalgamation bases in the category of inverse semigroups [14], and Renshaw introduced a homological structure in order to describe the amalgamated free product [32]. By using this structure, Renshaw proved that a semigroup U is an amalgmation base if, and only if, U has the extension property in every containing semi-group. Renshaw's result, which shows that a semigroup amalgam is embeddable if, and only if, it is embeddable as a monoid, allow us to focus on monoid amalgams. The subject of pomonoid amalgams was first studied by Fakhuruddin in 1986 but he only considered the commutative case [10]. Little work has been done in this category and recently Bulman-Fleming and Nasir revisited this area (see [7], [6], and [29]). They modified Fakhuruddin's definition of pomonoid amalgams, where they proved that a pomonoid amalgam that has the postrong representation extension property is strongly poembeddable [7]. They also proved that pogroups are strong poamalgamation bases in the category of pomonoids. Nasir [29] found that absolutely poatness pomonoids are strong poamalgamation bases in the category of commutative pomonoids. However, several questions remain unanswered in this area, and this research continues to study pomonoid amalgams by exploring when poembeddability can happen. It also aims to generalise some of the results in monoid amalgams. In addition, a number of subjects related to pomonoid amalgams have been considered, for example dominions and subpomonoid amalgams. New questions about the class of amalgamation bases have emerged recently and we briefly consider some of these