Prequantales and applications to semistar operations and module systems

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