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We show that a generalization of quantales and prequantales provides anoncommutative and nonassociative abstract ideal theoretic setting for thetheories of star operations, semistar operations, semiprime operations, idealsystems, and module systems, and conversely the latter theories motivate newresults on quantales and prequantales. Results include representation theoremsfor precoherent prequantales and multiplicative semilattices; acharacterization of the simple prequantales; and a generalization to thesetting of precoherent prequantales of the construction of the largest finitetype semistar operation and the largest stable semistar operation smaller thana given semistar operation.

Prequantales and applications to semistar operations and module systems