Semiring Orders in a Semiring

Given a semiring it is possible to associate a variety of partial orders with it in quite natural ways, connected with both its additive and its multiplicative structures. These partial orders are related among themselves in an interesting manner is no surprise therefore. Given particular types of semirings, e.g., commutative semirings, these relationships become even more strict. Finally, in terms of the arithmetic of semirings in general or of some special type the fact that certain pairs of elements are comparable in one of these orders may have computable and interesting consequences also. It is the purpose of this paper to consider all these aspects in some detail and to obtain several results as a consequence