## Resumo

The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, Z2-graded, superdivision

algebras (in a superdivision algebra each homogeneous element is invertible).

The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in Z2 × Z2-graded physics

(classical and quantum invariant models, parastatistics) we classify the associative Z2 × Z2-

graded superdivision algebras and show that 13 inequivalent cases have to be added to the

10-fold way. Our scheme is based on the “alphabetic presentation of Clifford algebras”, here

extended to graded superdivision algebras. The generators are expressed as equal-length

words in a 4-letter alphabet (the letters encode a basis of invertible 2 × 2 real matrices and

in each word the symbol of tensor product is skipped). The 13 inequivalent Z2 × Z2-graded

superdivision algebras are split into real series (4 subcases with 4 generators each), complex

series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators).