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We develop a non-classic algebraic theory for the purpose of investigating the convergence properties of dynamic routing protocols. The algebraic theory can be regarded as a generalization of shortest-path routing, where the new concept of free cycle generalizes that of a positive-length cycle. A primary result then states that routing protocols always converge, though not necessarily onto optimal paths, in networks where all cycles are free. Monotonicity and isotonicity are two algebraic properties that strengthen convergence results. Monotonicity implies protocol convergence in every network, and isotonicity assures convergence onto optimal paths. A great many applications arise as particular instances of the algebraic theory. In intra-domain routing, we show that routing protocols can be made to converge to shortest and widest paths, for example, but that the composite metric of Internet Gateway Routing Protocol (IGRP) does not lead to optimal paths. The more interesting applications, however, relate to inter-domain routing and its Border Gateway Protocol (BGP), where the algebraic framework provides a mathematical template for the specification, design, and verification of routing policies. We formulate existing guidelines for inter-domain routing in algebraic terms, propose new guidelines contemplating backup relationships between domains, and derive a sufficient condition for signaling correctness of internal-BGP.