In super case there are two possible definitions leading to nonisomorphic algebras (which are, however, isomorphic as vector spaces).

First definition. For a ℤ₂- graded vector space V = V₀⊕ V₁ define Λ (V): = T(V)/I, where T(V) is the tensor algebra of V and the ideal I is generated by Open image in new window for all v, w∈ V (tilde stands for parity). There is a natural ℤ-grading (degree) inherited from T(V). There is also a natural ℤ₂-grading (parity). The natural inclusion V→ Λ (V) is even (preserves parity) and maps elements of V to Λ¹ (V). The algebra Λ (V) is not commutative as a ℤ₂- graded algebra ; multiplication in it satisfies the condition Open image in new window for all elements v ∈ Λ ^k (V) and w∈ Λ ^l (V). Notice that degree and parity are independent. For a purely even V, parity is trivial (all elements of Λ (V) are even), and signs are due only to the degree of respective elements.

Second definition. For V = V₀⊕ V₁ consider the space with reversed parity π V, elements of which are denoted π v, where v∈ V (the canonical isomorphism (π V₀ = V₁, (π V)₁ = V₀ is written as the multiplication by an odd symbol π). Define E(V): = S(π V) = T(π V)/J, where the ideal J is generated by Open image in new window for all v, w ∈ V. By definition, the algebra E(V) is commutative (in ℤ₂-graded sense). The natural monomorphism V→ E(V), v ↦ π v, is odd (reverses parity). As well as for Λ (V), there is a natural ℤ-grading (degree), and the elements of V are mapped to E¹(V) = π V. Degree and parity are independent, except for a purely even or a purely odd V. For a purely even vector space V = V₀, the algebras Λ (V) and E(V) are isomorphic as algebras (not as ℤ₂- graded algebras , because in this case parity is trivial for Λ (V) and coincides with degree modulo 2 for E(V)). This is the classical exterior algebra . In general, the algebras Λ (V) and E(V) are not isomorphic. However, in every degree k, there is an isomorphism of vector spaces Λ ^k (V)≅ E ^k (V). The parity of this isomorphism is k mod 2. If we forget about ℤ₂-gradings and use the classical notions of exterior and symmetric algebras, then Λ ^k (V)≅ E ^k (V)≅ ⊕_{i+j = k}Λ ⁱ (V₀)⊗ S ^j (V₁). The difference in the multiplication laws in Λ (V) and E(V) is in the different commutation relation between (the images of) elements of V of opposite parity: in the case of Λ (V) their transposition gives the minus sign, and in the case of E(V) they strictly commute (no sign). The algebra E(V) has the advantage of being (super) commutative, so working with it seems more convenient (cf., however, [3].) The construction of E(V) was suggested by Bernstein and Leites [2]. Elements of E(V) can be regarded as polynomial functions on the superspace π V*. Here π V* is considered not as ℤ₂- graded vector space , but as the corresponding supermanifold . (Elements of E¹(V) = π V are linear functions on π V*.) Thus, E(V) can be naturally completed to Ê(V): = C^∞(π V*). The defining relations of the algebra Λ (V) are satisfied by the operators e(v) of the “exterior multiplication” by the elements of V on the space of Open image in new window [5].