ABSTRACT. The usual definition of regularity for convergence spaces can be characterized by a diagonal axiom R due to Cook and Fischer. The generalization of R to the realm of probabilistic convergence spaces depends on a t-norm T, and the resulting axiom Rr defines "T-regularity", which is the primary focus of this paper. We give several characterizations of T-regularity, both in general and for specific choices of T, and investigate some of its basic properties.
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