Some subset of 1. Ternary semigroups, groups and their bi-element representations. 2. Ternary Hopf algebras and analog of quantum Yang-Baxter equation. 3. Analog of universal enveloping algebra U_q(sl_2) with von Neumann regular commutative and noncommutative group-like generators, Von Neumann regular $R$-matrix, connection of with the crossed product, matrix representations, Verma moduli. 4. Von Neumann regular analogs of mappings, (super)manifolds, categories and functors, algebras and bialgebras. 5. Braid semistatistics, von Neumann regular analog of braiding and doubly regular Hopf algebras. 6. Von Neumann regular set-theoretical and constant solutions of quantum Yang-Baxter equations over Grassmann algebras, 6-vertex solutions, von Neumann regular analog of quantum plane. -------------------