mathematics

Ternary commutator

Ternary commutator is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ternary commutator rather than just read about it. In short: In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by [ a , b , c ] = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle [a,b,c]=abc-acb-bac+bca+cab-cba.\,} Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, whereas the 2-commutator is the ordinary commutator. Properties When one or more of…

Key takeaways

Ternary commutator belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ternary commutator to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ternary commutator from memory before moving on to harder problems.

Reference excerpt

In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by

[ a , b , c ] = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle [a,b,c]=abc-acb-bac+bca+cab-cba.\,}

Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, whereas the 2-commutator is the ordinary commutator.

Properties When one or more of a, b, c is equal to 0, [a, b, c] is also 0. This statement makes 0 the absorbing element of the ternary commutator. The same happens when a = b = c.

Further reading Bremner, Murray R. (15 August 1998), "Identities for the Ternary Commutator", Journal of Algebra, 206 (2): 615–623, doi:10.1006/jabr.1998.7433 Bremner, Murray R.; Ortega, Juana Sánchez (25 October 2010), "The partially alternating ternary sum in an associative dialgebra", Journal of Physics A: Mathematical and Theoretical, 43 (56) 455215, arXiv:1008.2721, Bibcode:2010JPhA...43S5215B, doi:10.1088/1751-8113/43/45/455215, S2CID 6636902 Bremner, Murray R.; Peresi, Luiz A. (1 April 2006), "Ternary analogues of Lie and Malcev algebras", Linear Algebra and Its Applications, 414 (1): 1–18, doi:10.1016/j.laa.2005.09.004 Bremner, Murray R.; Peresi, Luiz A. (26 July 2012), "Higher identities for the ternary commutator", Journal of Physics A: Mathematical and General, 45 (50) 505201, arXiv:1207.6312, Bibcode:2012JPhA...45X5201B, doi:10.1088/1751-8113/45/50/505201, S2CID 17037773 Devchand, Chandrashekar; Fairlie, David; Nuyts, Jean; Weingart, Gregor (6 November 2009), "Ternutator identities", Journal of Physics A: Mathematical and Theoretical, 42 (47) 475209, arXiv:0908.1738, Bibcode:2009JPhA...42U5209D, doi:10.1088/1751-8113/42/47/475209, S2CID 17246666 Nambu, Yoichiro (1973), "Generalized Hamiltonian Dynamics", Physical Review D, 7 (8): 2405–2412, Bibcode:1973PhRvD...7.2405N, doi:10.1103/PhysRevD.7.2405

Worked examples

Example 1 — a first encounter with Ternary commutator

Start with the simplest possible case. Write down what Ternary commutator claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ternary commutator before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ternary commutator ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ternary commutator

In research: Ternary commutator appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ternary commutator in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ternary commutator is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstract algebra stubs, Algebras, Ternary operations, so understanding it makes those chapters shorter.
In everyday life: Look for Ternary commutator outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Ternary commutator in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ternary commutator means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ternary commutator out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ternary commutator in simple terms?

In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by [ a , b , c ] = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle [a,b,c]=abc-acb-bac+bca+cab-cba.\,} Also called the ternutator or alternating ternary sum, it is a special…

Why does Ternary commutator matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ternary commutator?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ternary commutator.