Congruences Over An Additively Idempotent Commutative Semiring

Congruences play a great role in the study of semirings.This dissertation uses prime congruences and twisted-prime congruences over a semiring to investigate the basic properties of congruences over an additively idempotent commutative semiring.The main results are as follows:1.The characterizations of congruences generated by arbitrary relation over an additively idempotent commutative semiring and the properties of prime congruences and twist-prime congruences over an additively idempotent commutative semiring are given.2.The conditions under which the annihilators of the elements in an additively idempotent commutative semiring being prime congruences or twistedprime congruences are given.The properties of the elements in the set of multiplicative generalized powers and relationship between the elements and the annihilators are given.3.A topological space on the direct product of the tropical semiring are established.The conditions under which the congruences over the tropical polynomial semiring twisted-prime congruences or flat congruences are given.The smallest prime congruence over the tropical polynomial semiring is not twistedprime congruence and is intersection decomposable.