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This paper considers a class of nonsymmetric cooperative neural networks (NNs) where the neurons are fully interconnected and the neuron activations are modeled by piecewise linear (PL) functions. The solution semiflow generated by cooperative PLNNs is monotone but, due to the horizontal segments in the neuron activations, is not eventually strongly monotone (ESM). The main result in this paper is that it is possible to prove a peculiar form of the Limit Set Dichotomy for this class of cooperative PLNNs. Such a form is slightly weaker than the standard form valid for ESM semiflows, but this notwithstanding it permits to establish a result on convergence analogous to that valid for ESM semiflows. Namely, for almost every choice of the initial conditions, each solution of a fully interconnected cooperative PLNN converges toward an equilibrium point, depending on the initial conditions, as t \to +\infty. From a methodological viewpoint, this paper extends some basic techniques and tools valid for ESM semiflows, in order that they can be applied to the monotone semiflows generated by the considered class of cooperative PLNNs. [ABSTRACT FROM PUBLISHER]