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Nuclei and applications to star, semistar, and semiprime operations

We show that the theory of quantales and quantic nuclei motivate new results on star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results on quantales and quantic nuclei. Results include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of the simple prequantales; and a generalization to the setting of precoherent quantales of the construction of the largest finite type semistar operation and the largest stable semistar operation smaller than a given semistar operation.