Definition 1

Given the above and a commutation factor $\varepsilon$ on $\Delta \times \Delta$, an $\varepsilon$-derivation (or $(K, \varepsilon)$-derivation) of degree $\delta \in \Delta$ of $(A, A', A'')$ into $(B, B', B'')$ is a triple of graded $K$-linear mappings of degree $\delta$:

such that, for every homogeneous element $a \in A$ and every element $a' \in A'$ (4)

It obviously suffices by linearity to verify relation (4) when $a$ and $a'$ run through respective generating systems of $A$ and $A'$.

It is often convenient to denote the three mappings $d, d', d''$ by the same letter $d$ (which can be justified by denoting equally by $d$ the graded $K$-linear mapping of degree $\delta$)

of $A \oplus A' \oplus A''$ into $B \oplus B' \oplus B''$). Relation (4) can then be written more simply

The $\varepsilon$-derivations of $(A, A', A'')$ into $(B, B', B'')$ of given degree form a sub-K-module of the K-module of graded linear mappings When $\varepsilon(\alpha, \beta) = 1$ for all $\alpha, \beta$ in $\Delta$, we say simply derivation (or K-derivation) instead of $\varepsilon$-derivation. Derivations form a sub-K-module of When $\Delta = \mathbf{Z}$ and $\varepsilon(p.q) = (-1)^{pq}$, every $\varepsilon$-derivation of even degree is a derivation; every $\varepsilon$-derivation of odd degree is often called an antiderivation (or K-antiderivation); an antiderivation $d$ therefore satisfies for a homogeneous element $a \in A$.