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Matroids over skew tracts provide an algebraic framework simultaneouslygeneralizing the notions of linear subspaces, matroids, oriented matroids,phased matroids, and some other "matroids with extra structure". Asingle-element extension of a matroid $\mathcal{M}$ over a skew tract $T$ is amatroid $\widetilde{\mathcal{M}}$ over $T$ obtained from $\mathcal{M}$ byadding one more element. Crapo characterized single-element extensions ofordinary matroids, and Las Vergnas characterized single-element extensions oforiented matroids, in terms of single-element extensions of their rank 2contractions. The results of Crapo and Las Vergnas do not generalize tomatroids over skew tracts, but we will show a necessary and sufficientcondition on skew tracts, called Pathetic Cancellation, such that the resultcan generalize to weak matroids over skew tracts.  Stringent skew hyperfields are a special case of skew tracts which behave inmany ways like skew fields. We find a characterization of single-elementextensions of strong matroids over stringent skew hyperfields.

Extensions of weak matroids over skew tracts and strong matroids over stringent skew hyperfields