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Suppose I have a ring $(A, ◦, •)$ where $A$ is a set of elements $\{α, β, γ,\ldots\}$.

Can $◦$ and $•$ with which the ring is equipped be properly termed, in English, its "internal laws of composition," or is "internal binary operators" better in this context?

Are $A$, $◦$, and $•$ taken together—when written as $(A, ◦, •)$, e.g.—properly termed a ternary, triad, or triplet, in English?

I am trying to translate the Spanish:

Se denomina anillo a una terna $(A, ◦, •)$, donde $A$ es un conjunto de elementos $\{α, β, γ, …\}$, dotado de dos leyes de composición internas, $◦$ y $•$, …

thank you

Rodrigo de Azevedo
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Geremia
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1 Answers1

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I would translate it as:

"A ring is a triplet $(A, \circ, \bullet)$, where $A$ is the set $\{\alpha, \beta, \gamma, \ldots, \}$ with two binary operations $\circ$ and $\bullet$".

This is not a literal translation, but the literal translation would be a bit awkward in my opinion.

As is clear now, I choose to translate 'internal laws of composition' as 'binary operations'.

'Internal binary operations' is uncommon and redundant since the definition of binary operation entails that it is 'internal'.

Git Gud
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  • Yes, I didn't realize that the literal translation is an older way of saying "binary operation". Why "triplet" and not "triad"? And why not "internal binary operation" instead of just "binary operation"? thanks – Geremia Mar 15 '14 at 23:26
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    Simply because it is more common (I think), I've never seen triad. As for the second question, my answer already answers it: it's redudant and uncommon. I also have never seen it. – Git Gud Mar 15 '14 at 23:27
  • How is it redundant? See this. The quote I gave is introducing rings, not assuming the reader knows anything about them. – Geremia Mar 15 '14 at 23:30
  • Also, isn't "operator" slightly better than "operation"? – Geremia Mar 15 '14 at 23:33
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    @Geremia I misunderstood the concept of 'internal', but I mantain my translation. If one wants to be very careful, one could add "with two binary operations $\circ$ and $\bullet$ defined on $A$" and by definition of binary operation, they are internal operations. I should also add that I've never seen these concepts of internal and external operations in english. – Git Gud Mar 15 '14 at 23:34
  • Just for the record: the expression "internal composition law" is currently used in french: http://fr.wikipedia.org/wiki/Loi_de_composition_interne and until now I was thinking it was the same in english... We, frenchies, loves all those baroque embellishments :p ! – Sebastien Mar 15 '14 at 23:35
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    @Geremia Operation is an established term. Operator is too, but with a different meaning, operator usually is a function that takes functions as inputs. – Git Gud Mar 15 '14 at 23:36
  • @Sebastien: Is that the same as saying it's closed? – Geremia Mar 15 '14 at 23:45
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    Also, it seems "external binary operation" is proper English, but for some reason not "internal binary operation"? – Geremia Mar 16 '14 at 00:01
  • @Geremia Seems so, but you're only interested in internal binary operations and in this case binary operation is much more common. – Git Gud Mar 16 '14 at 00:02
  • It's hard because saying "defined on $A$" is only half of what "internal" means… I suppose I could say it's "defined on $A$" and closed? Ah, but it's so much easier to just say "internal"… – Geremia Mar 16 '14 at 00:08
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    @Geremia No, being closed means that the outputs are elements of $A$. I want to take back what I said about them being defined on $A$, they are actually defined on $A^2$. What I meant to say was: "with two binary operations $∘$ and $∙$ on A". There is no risk of confusion because of the definition of binary operation. – Git Gud Mar 16 '14 at 00:10
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    @Geremia: if "closed" means "the output is in A" then yes, "interne" in "loi de composition interne" holds for "closed". More precesily, a "loi de composition interne" on a set $E$ is simply an application $E\times E \to E$. That's it (which is the same as "binary operation" in english). Damn! Vocabulary in english is far more intuitive than in french ! – Sebastien Mar 16 '14 at 08:38
  • @Geremia Do you still require help with this? – Git Gud Mar 28 '14 at 23:48
  • @GitGud: No, but thanks for reminding me to mark your answer as the answer. ☺ – Geremia Mar 29 '14 at 01:17
  • @Geremia Thank you for the '$\huge ☺$' thing. – Git Gud Mar 29 '14 at 10:36
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    @GitGud: You're welcome. It's UTF-16 0x263A. There's a frowning one, too: $\huge{☹}$ (0x2639), a postal mark one: $\huge{〠}$ (0x3020), and a black smiling one: $\huge{☻}$ (0x263B). – Geremia Mar 29 '14 at 15:11