NonCommutativeMultiply

The ordered product, for units that anticommute. $\times$ cannot host one: Multiply is declared commutative, so canonicalisation sorts its operands before any handler runs and the sign is lost.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageNonCommutativeMultiply!↗

NonCommutativeMultiply(a, b, …)the product of the operands, in the order written

GeometricProduct(a, b, …)alias, for the geometric-algebra reading

a \otimes binfix alias ($\otimes$)

Typed signatures

Type	Package	Applies
untyped	hypercomplex	always
(number*) -> number	structures	always

Domain: Structures

Details

• Wolfram spells this head the same way (infix **), keeping it apart from Times; matrix multiplication is likewise its own head, Dot
• On the COMMUTING families it simply agrees with $\times$, so it is safe to use everywhere
• A tensor product of two scalars is their product, so $2 \otimes 3$ is $6$
• Associative across mixed families: the commutation factor $\varepsilon(g,h) = (-1)^{\mathrm{anti}(g)\mathrm{anti}(h)}$ is a bicharacter
• Juxtaposition carries the sign on its own -- $e_2e_1$ is caught before Multiply's commutative sort can reach it
• An explicit $\times$ or $\cdot$ is not: it parses straight to a sorted Multiply, so it REFUSES two distinct anticommuting units rather than assert a sign it cannot justify

Examples

Basic #

the canonical order

live ↗

In$\mathrm{NonCommutativeMultiply}(e_1, e_2)$Out$e_1e_2$

Symbolic products are held, and identical ones compare equal

live ↗

In$\mathrm{NonCommutativeMultiply}(a, b, c)=\mathrm{NonCommutativeMultiply}(a, b, c)$Out$\mathrm{True}$

live ↗

In$\begin{cases}ab=ab\\\mathrm{NonCommutativeMultiply}(a, b)=\mathrm{NonCommutativeMultiply}(b, a)\end{cases}$Out$\bigl\lbrack\mathrm{True}, \mathrm{NonCommutativeMultiply}(a, b)=\mathrm{NonCommutativeMultiply}(b, a)\bigr\rbrack$

Scope #

scalars: the implied identity factor

live ↗

In$2⊗3$Out$6$

commuting units: agrees with $\times$

live ↗

In$i_2⊗i_1$Out$i_1i_2$

Applications #

the quaternion product $(1+2i+3j+4k)(2-i+j-3k) = 13-10i+9j+10k$, with $i = f_1$, $j = f_2$, $k = f_1f_2$

live ↗

In$\mathrm{NonCommutativeMultiply}(1+2f_1+3f_2+4f_1f_2, 2-f_1+f_2-3f_1f_2)$Out$10f_1f_2-10f_1+9f_2+13$

Properties #

reversing the operands flips the sign

live ↗

In$\mathrm{NonCommutativeMultiply}(e_2, e_1)$Out$-(e_1e_2)$

$(e_1e_2)^2 = -1$

live ↗

In$\mathrm{NonCommutativeMultiply}(e_1, e_2, e_1, e_2)$Out$-1$

associative: nested products flatten

live ↗

In$\mathrm{NonCommutativeMultiply}(a, \mathrm{NonCommutativeMultiply}(b, c))$Out$\mathrm{NonCommutativeMultiply}(a, b, c)$

undeclared symbols aren't assumed to commute, so the order is kept

live ↗

In$\mathrm{NonCommutativeMultiply}(b, a)$Out$\mathrm{NonCommutativeMultiply}(b, a)$

live ↗

In$\bigl\lbrack\mathrm{NonCommutativeMultiply}(0, a), \mathrm{NonCommutativeMultiply}(1, a)\bigr\rbrack$Out$\bigl\lbrack\mathrm{NonCommutativeMultiply}(0, a), \mathrm{NonCommutativeMultiply}(1, a)\bigr\rbrack$

live ↗

In$\mathrm{NonCommutativeMultiply}(a+b, c)-ac$Out$\mathrm{NonCommutativeMultiply}(a+b, c)-ac$

Possible issues #

plain $\times$ declines: both orders arrive identical

live ↗

In$e_2e_1$Out$e_1e_2$

A single factor is just itself

live ↗

In$\mathrm{NonCommutativeMultiply}(a)$Out$a$

About

Examples	14
Checked against	Wolfram Language 8 agree2 differ4 unevaluatedSageMath 6 agreeSymPy 1 agree
Crosswalk	Wolfram Language

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: Norm, Basis, Quaternions