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AbstractHara and Yoshida introduced a notion of α-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from thedoi:10.1017/s0027763000025952 fatcat:w7lkzexninfejedi6iegfqotwu