Question about Algebraic structure?

Question

Mathematicians use parentheses to represent a algebraic structure e.g. $ (G, \times)$. Also parentheses are used to represent ordered pair (or general ordered n-tuple) e.g. $(x,y)$.
My question: Is algebraic structure in fact a ordered pair consisting of set and operation?

I have friend, and he says Yes, but I am little suspicions. Thanks.

What other way would you want to define algebraic structure? — Git Gud, Dec 07 '13 at 18:06
I don't want other way. Is there any? And what dose it mean to write $(\times,G)$, or any other combination where set is not on first place? Thanks, once again, for your responses. — , Dec 07 '13 at 18:15
I've never seen anything other than 'the' set in the first entry of the order pair, triplet or whatever. — Git Gud, Dec 07 '13 at 18:17

Robert Lewis · Accepted Answer · 2013-12-07T18:35:15.097

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The situation may perhaps be clarified if one recalls that functions are often defined as sets of ordered pairs. That is, a function $f:X \to Y$ is a subset of $X \color{red}\times Y$ with the additional property that if $(x, y_1), (x, y_2) \in f \subset X \color{red}\times Y$ then we must have $y_1 = y_2$; this latter requirement formalizing the notion that $f$ takes a unique value for each $x$, as opposed to a mere relation which may have $(x, y_i) \in f$ for many $y_i \in Y$. Thus, in the case of "$\times$" in $(G, \times)$ being a binary operation, it represents a set of ordered pairs in $G \color{red}\times G$; we might even write $\times \subset G \color{red}\times G$, although care must be taken to distinguish between the two meanings of the symbols "$\times$" in this formal "sentence". In any event, the ordered pair $(G, \times)$ has as its first element the set $G$, and as its second the set $\times \subset G \color{red}\times G$; it is an ordered pair of sets. I trust the potentially confusing use of the symbol $\times$ here is nevertheless clear enough to be unambiguous; the context and color should resolve any doubts as to which "$\times$" means what!

Note added edit: At the suggestion of our colleague Git Gud (see his comments below), I have use the red typeface for $\times$ as Cartesian product of two sets. This should further reduce any chance of confusion. End note.

Hope this helps. Cheers,

and as always,

Fiat Lux!!!

edited Dec 07 '13 at 18:35

answered Dec 07 '13 at 18:21

Robert Lewis

71,180

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Why not write one of the $\times$ with a different color? – Git Gud Dec 07 '13 at 18:22
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Damn fine idea! How? I haven't sussed out colored text in Latex yet! – Robert Lewis Dec 07 '13 at 18:24
Or use other simbol for operation? – Dec 07 '13 at 18:24
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$\color{blue}{1+1=2}$ yields $\color{blue}{1+1=2}$, it also works with red, green, yellow and most likely a lot others. – Git Gud Dec 07 '13 at 18:24
Thanks, let me do a tad of editing to try this out! – Robert Lewis Dec 07 '13 at 18:26
This is a great answer. I linked to it in similar question I created: http://math.stackexchange.com/questions/964206/algebraic-structures-does-order-matter/964216#964216 – EthanAlvaree Oct 08 '14 at 20:13
@Mathemanic: Your good word are warmly and well received; many thanks. And thanks for the link to another interesting question on a similar topic. Regards! – Robert Lewis Oct 09 '14 at 04:29

score 1 · Answer 2 · answered Dec 07 '13 at 18:48

The answer to your question is sort-of.

One way to think about algebraic structures is that they consist of a set of elements, and there are (possibly) some operations you can apply to elements.

Thus, when we think this way, we need some sort of data structure to hold all of the relevant pieces if we want to work with them in a set-theoretic manner.

For example, for groups, we need a data structure that contains the set of elements in the group, the multiplication operation, the inversion operation, and which element is the identity.

The standard way to create this data structure is to use an ordered 4-tuple: the first entry contains the set of elements, the second contains the identity element, the third contains the inversion operation, and the fourth contains the multiplication operation. The operations are stored in this 4-tuple in the form of set-theoretic functions.

Other data structures could be used as well to denote groups, so long as these four pieces of information can still be extracted. An ordered pair consisting of just the set of elements and the multiplication operation is a popular choice.

This convention also gives a cute notational shortcut: saying "the group $(G, 0, -, +)$" is much more concise than "a group whose set of elements we will call $G$, whose group operation we will denote as $+$, whose negation element we will denote as $-$, and whose identity we will write as $0$".

Question about Algebraic structure?

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