Grassmann number

An element of an exterior algebra generated by anticommuting variables, with odd generators squaring to zero.

Version

v1 · 2026-09-08 · History

Domain-specific #

4774

Origin domain

mathematical physics

Subdomain

mathematical physics

Aliases

Anticommuting number, Supernumber

Core Idea

A general Grassmann number can contain even and odd graded components, it is not an ordinary complex number despite complex coefficients, and anticommutation applies with graded parity rather than to every pair of arbitrary elements. Formal generators obey theta_i theta_j equals minus theta_j theta_i, forcing repeated odd factors to vanish; finite polynomial combinations form a Z2-graded algebra with Berezin differentiation and integration rules. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Grassmann number belongs to mathematical physics and is useful where the analyst can specify the typed mathematical physics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the base field and generating vector space, exterior algebra, anticommuting generators, nilpotence of individual generators, ordered monomial basis, even and odd parity grading, body and soul decomposition when used, multiplication sign rule, conjugation convention and Grassmann differentiation or integration context are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the base field and generating vector space, exterior algebra, anticommuting generators, nilpotence of individual generators, ordered monomial basis, even and odd parity grading, body and soul decomposition when used, multiplication sign rule, conjugation convention and Grassmann differentiation or integration context are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Grassmann number. Grassmann number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed mathematical physics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the base field and generating vector space, exterior algebra, anticommuting generators, nilpotence of individual generators, ordered monomial basis, even and odd parity grading, body and soul decomposition when used, multiplication sign rule, conjugation convention and Grassmann differentiation or integration context are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of mathematical physics because they reuse the typed mathematical physics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Formal generators obey theta_i theta_j equals minus theta_j theta_i, forcing repeated odd factors to vanish; finite polynomial combinations form a Z2-graded algebra with Berezin differentiation and integration rules., and type the carrier, state every parameter and convention in the definition, test that the base field and generating vector space, exterior algebra, anticommuting generators, nilpotence of individual generators, ordered monomial basis, even and odd parity grading, body and soul decomposition when used, multiplication sign rule, conjugation convention and Grassmann differentiation or integration context are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Grassmann number Domain-specific

Parents (1) — more general patterns this builds on

• Grassmann number is a kind of Formalization Prime

  The proposed strict upward parent is prime:formalization.

Hierarchy paths (2) — routes to 2 parentless roots

• Grassmann number → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space

Grassmann number sits in a moderately populated region (41^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebras, Quantization & Operators (17 abstractions)

Nearest neighbors

• Spin network — 0.90
• Geometric quantization — 0.90
• Tensor field — 0.89
• Fundamental unit (number theory) — 0.89
• De Donder–Weyl theory — 0.89

Computed from structural-signature embeddings · 2026-09-08