Congruence-free commutative semirings

Abstract

By a commutative semiring we mean a semiring in which both addition and multiplication are commutative. A semiring S is congruence-free iff the only congruences on S are S × S and the identity congruence. In this thesis we characterize congruence-free commutative semirings with a multiplicative identity, S as follows: Theorem: If S has a multiplicative zero which is also an additive identity then S is a field or a semifield of order 2. Theorem: If S has a multiplicative zero which is also an additive zero then S is a semifield. Theorem:There exist such semirings S which have no multiplicative zeros which are not division semirings. Theorem: If S has no multiplicative zero then either S is a band with respect to addition or S is additively cancellative. Theorem:If S has no multiplicative zero and S is additively cancellative then S has a natural partial order ≥ . If ≥ is total, then S is a division semiring.