Coherent continuous systems and the generalized functional equation of associativity.

*(English)*Zbl 0648.39006This paper is a contribution to the development of a theory of coherent continuous systems. Binary systems are concerned with only two states (working or failed), while continuous systems can be used to model gradual degradation of component performance. These theories serve as foundation for the study of reliability of systems.

The author introduces the idea of a coherent continuous system, which means that the structure function of the system is continuous and satisfies some monotonicity and non-degeneracy conditions. In order to deduce the possible forms of such structure functions, modular decomposition is then discussed. This follows the original work of Z. W. Birnbaum and J. D. Esary [J. Soc. industr. appl. Math. 13, 444-462 (1965; Zbl 0235.94029)] on binary systems and leads to the generalized functional equation of associativity \((GFEA)\quad G(g(u,v),w)=H(u,h(v,w))\) for continuous functions G, g, H, h satisfying the monotonicity and non-degeneracy conditions inherited from the structure function. The author further imposes some technical assumptions in the form of boundary conditions. These conditions are not well- motivated from the standpoint of the theory of reliability of systems, but they do allow the author to obtain desirable structure functions. Moreover, some of these structure functions are not found under other conditions which have been used to solve (GFEA): for instance cancellativity conditions used by J. Aczél [Adv. Math. 1, 383-450 (1965; Zbl 0135.036)] and M. A. Taylor [Aequationes math. 17, 154- 163 (1978; Zbl 0405.39009)], and strict monotonicity used by T. C. Koopmans [Studies math. managerial Economics 12, 57-78 (1972; Zbl 0253.90003)]. In order to achieve his main result, the author uses results about threads from the theory of topological semigroups.

The author introduces the idea of a coherent continuous system, which means that the structure function of the system is continuous and satisfies some monotonicity and non-degeneracy conditions. In order to deduce the possible forms of such structure functions, modular decomposition is then discussed. This follows the original work of Z. W. Birnbaum and J. D. Esary [J. Soc. industr. appl. Math. 13, 444-462 (1965; Zbl 0235.94029)] on binary systems and leads to the generalized functional equation of associativity \((GFEA)\quad G(g(u,v),w)=H(u,h(v,w))\) for continuous functions G, g, H, h satisfying the monotonicity and non-degeneracy conditions inherited from the structure function. The author further imposes some technical assumptions in the form of boundary conditions. These conditions are not well- motivated from the standpoint of the theory of reliability of systems, but they do allow the author to obtain desirable structure functions. Moreover, some of these structure functions are not found under other conditions which have been used to solve (GFEA): for instance cancellativity conditions used by J. Aczél [Adv. Math. 1, 383-450 (1965; Zbl 0135.036)] and M. A. Taylor [Aequationes math. 17, 154- 163 (1978; Zbl 0405.39009)], and strict monotonicity used by T. C. Koopmans [Studies math. managerial Economics 12, 57-78 (1972; Zbl 0253.90003)]. In order to achieve his main result, the author uses results about threads from the theory of topological semigroups.

Reviewer: B.Ebanks

##### MSC:

39B99 | Functional equations and inequalities |