Algebraic structures spanned by differential-like operators on toy Fock spaces

We study multiplicative systems of linear mappings acting on the toy Fock space, a.k.a.\ Rademacher chaos or Walsh-Fourier series, related to the creation, annihilation, and conservation operators in quantum probability. Like differential operators they entail analogs of the Leibnitz Formula and the Chain Rule, derived with the help of Riesz products (or discrete coherent vectors). Two symmetries among these operators entail groups and algebras of varying signatures and mixed commutativity, generalizing Pauli spin matrices and quaternions. In particular, anticommutative Rademacher systems are constructed.