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In this paper, the concept of a T -partition is introduced as a generalization of that of a classical partition. The approach is based on the observation that for any two members of a classical semi-partition, the nonemptiness of their intersection implies their equality. This observation is generalized to T -semi-partitions using degrees of compatibility and equality based on a t-norm. T and its biresidual operator E T By imposing an additional covering condition, the concept of a T -partition is obtained. An interesting numerical characterization of T -partitions is proved, leading to a desired one-to-one correspondence between T -partitions and T -equivalences. Moreover, the refinement of T -partitions is discussed. In particular, it is shown that the T ^∗-refinement of any two T -partitions of a given universe is again a T -partition of that universe if and only if the t-norm T ^∗ dominates the t-norm T .