In [13,14,7], the modeling of silent tasks by means of so-called r-operators has been studied, and interesting relations have been shown between algebraic properties of a given operator and stabilizing properties of the related distributed algorithms. Modeling algorithms with algebraic operators allows to determine generic results for a wide set of distributed algorithms. Moreover, by simply checking some local algebraic properties, some global properties can be deduced. Stabilizing properties of shortest path calculus, depth-first-search tree construction, best reliable transmitters, best capacity paths, ordered ancestors list... have hence been established by simply reusing generic proofs, either in the read-write shared register models [13,14] or in the unreliable message passing models [7]. However, while this approach is promising, it may be penalized by the difficulty in designing new r-operators. In this paper, we present the fundation of the r-operators by introducing a generalization of the idempotent semi-groups, called r-semi-group. We establish the requirements on the operators to be used in distributed computation and we show that the r-semi-groups fulfill them. We investigate the connections between semi-groups and r-semi-groups, in order to ease the design of r-operators. We then show how to build new r-operators, to solve new algorithmic problems. With these new results, the r-semi-groups appear to be a powerful tool to design stabilizing silent tasks.
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