anticommutative
|an-ti-com-mu-ta-tive|
/ˌæn.ti.kəˈmjuː.tə.tɪv/
against commutativity (order matters; sometimes sign flips)
Etymology
'anticommutative' originates from Modern English, specifically combining the prefix 'anti-' (from Greek 'anti' meaning 'against' or 'opposite') and the adjective 'commutative' (from Latin 'commutare' meaning 'to change or exchange').
'commutative' comes from Latin 'commutare' ('com-' meaning 'together' + 'mutare' meaning 'to change'), passed through Late Latin and Middle French into English as 'commute/commutative'; the prefix 'anti-' entered English from Greek via Latin and Old French, and the compound 'anticommutative' formed in Modern English by prefixing 'anti-' to 'commutative'.
Initially the parts meant 'against' + 'change/exchange'; combined, the term came to mean 'against commutativity' or 'not commutative', and in many mathematical contexts it was specialized to mean 'sign-reversing under swap' (a*b = −b*a).
Meanings by Part of Speech
Adjective 1
not commutative; indicating that the order of operands matters because swapping them changes the result (i.e., a*b ≠ b*a in general).
Matrix multiplication is anticommutative in the sense that A B often differs from B A; however, mathematicians more commonly call this noncommutative.
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Adjective 2
having the property that swapping operands reverses the sign: a*b = − b*a. This usage appears in algebra (e.g., cross product, Lie bracket, exterior product) and is often called skew-commutative or anti-commutative.
The vector cross product is anticommutative because u × v = −(v × u).
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Last updated: 2025/08/29 19:41