Partial orders and idempotents of monoids

Idempotents of the monoid play different roles in the formation of its properties. A set of idempotents is divided into three parts: incomparable with a unit, less and equal to a unit, and more and equal to a unit. The idempotents of the first part are called primary and the idempotents comparable with a unit are called secondary. The properties of idempotents are investigated in terms of partial orders and Green’s equivalences. In the article the main attention is given to finding connections among different classical and non-classical, stable and unstable partial orders and roles which the idempotents play in that. In particular, as a result, the criterion of stability of Mitsch’s partial order is obtained. Different examples of ordered monoids are shown in the context of the constructed theory of idempotents and partial orders.