Within the context of an algebraic theory of processes, an equational specification of process cooperation is provided. Four cases are considered: free merge or interleaving, merging with communication, merging with mutual exclusion of tight regions, and synchronous process cooperation. The rewrite system behind the communication algebra is shown to be confluent and terminating (modulo its permutative reductions). Further, some relationships are shown to hold between the four concepts of merging.

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We present an axiom system ACP, for communicating processes with silent actions ('z-steps'). The system is an extension of ACP, Algebra of Communicating Processes, with Milner's z-laws and an explicit abstraction operator. By means of a model of finite acyclic process graphs for ACP, syntactic properties such as consistency and conservativity over ACP are proved. Furthermore, the Expansion Theorem for ACP is shown to carry over to ACP~. Finally, termination of rewriting terms according to the ACP~ axioms is proved using the method of recursive path orderings.

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0 −1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system? ACM Reference Format: Bergstra, J. A. and Tucker, J. V. 2007. The rational numbers as an abstract data type

Abstract. We take a thread as the behavior of a sequential deterministic program under execution and multi-threading as the form of concurrency provided by contemporary programming languages such as Java and C#. We outline an algebraic theory about threads and multi-threading. In the case of multi-threading, some deterministic interleaving strategy determines how threads are interleaved. Interleaving operators for a number of plausible interleaving strategies are specified in a simple and concise way. By that, we show that it is essentially open-ended what counts as an interleaving strategy. We use deadlock freedom as an example to show that there are properties of multi-threaded programs that depend on the interleaving strategy used.