Totally positive shapes and TPk-completable patterns

The notions of total positivity and of TPk are generalized to “shapes” (a generalization of matrices). In particular, the relationship between positivity of “contiguous” minors and all minors is characterized for general shapes and for certain special types of shapes. This and other ideas are used to address the TPk-completion problem and TPk-completable patterns. In case k=2, a near characterization of TP2-completable patterns is given.