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Every finite-dimensional algebra over an infinite field has a basis consisting of units. If the algebra is interior for a finite p-group D, we say that it is a bipermutation D-algebra if there is a basis that is invariant under left and right D-multiplication. Such a basis may be viewed as a (D, D)-biset, and if the basis can be chosen to consist of units, it (almost) determines a saturated fusion system on D. This talk, based on joint work with Laurence Barker, will have two goals: To explain the material in the above paragraph in greater detail, and to show how certain structural properties of an algebra are equivalent to the existence of an unital biinvariant basis. The ultimate goal--applying these results to the source algebra of a block--will be outlined at the end, time permitting.