anti-commutative
|an-ti-com-mu-ta-tive|
/ˌæn.ti kəˈmjuː.tə.tɪv/
order matters (may reverse sign)
Etymology
'anti-commutative' is a modern English compound formed from the prefix 'anti-' (from Greek anti- meaning 'against' or 'opposite') and 'commutative' (relating to 'commute').
'commutative' traces back to Latin 'commutare' (com- 'together' + mutare 'to change'); the mathematical sense 'commutative' (operation where order can be exchanged) developed in modern mathematics, and 'anti-' was later attached to denote the opposite or negation of that property.
Initially, elements of the compound referred simply to being 'against' or 'opposite' the notion of being able to 'commute' (exchange) without change; over time it acquired the specific mathematical senses 'not commutative' and the narrower algebraic sense 'swapping reverses sign.'
Meanings by Part of Speech
Noun 1
the property of being anti-commutative; (as a noun) 'anti-commutativity' denotes that an operation or relation is anti-commutative.
The anti-commutativity of the bracket is essential to the structure of the Lie algebra.
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Adjective 1
not commutative; the result depends on the order of the operands (a * b ≠ b * a).
Matrix multiplication is anti-commutative in this context: A B is not equal to B A.
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Adjective 2
specifically in algebra, skew-commutative or anti-commutative in the narrow sense: swapping two elements reverses the sign (a * b = − b * a).
The Lie bracket [x, y] is anti-commutative because [x, y] = −[y, x].
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Last updated: 2025/10/22 06:57