  Five algebraic notions of termination are formalised, analysed and compared:
wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration,
absence of divergence and normalisation. The study is based on modal semirings,
which are additively idempotent semirings with forward and backward modal
operators. To model infinite behaviours, idempotent semirings are extended to
divergence semirings, divergence Kleene algebras and omega algebras. The
resulting notions and techniques are used in calculational proofs of classical
theorems of rewriting theory. These applications show that modal semirings are
powerful tools for reasoning algebraically about the finite and infinite
dynamics of programs and transition systems.
