Group theory

[Anonymous No.9463508]

Can a group be closed under more than one operation?

[Anonymous No.9463515]

>>9463508
Wrong terminology my man. When talking of a group, a single operation is implied. And it is already implied that the set in the group structure is closed by it. The correct way of asking your question, if I understood you correctly, is "Can a set be closed under more than one operation?"

[檢查人員:性感的腳步 No.9463593]

>>9463515
Welcome to Infinity. (Translation delay = \Pi - \Phi + 1 = 1/2)

[Anonymous No.9463613]

>>9463593

I'm not sure why, but you really bast my ass. I'm usually immune to obvious trolling, but this is really making me butt bludgeoned. Good job.

[Anonymous No.9463619]

>>9463508
A field is a group under both addition and multiplication.

[檢查人員:性感的腳步 No.9463728]

>>9463613
他們是誰?

[Anonymous No.9463785]

>>9463728

我说中文. Well, not really. Just a little. And I'm pretty sure 他們 is plural. It should just be 他是誰. You've just outted yourself as using google translate and not being a real chinese speaker.

[Anonymous No.9463797]

>>9463619
Incorrect

[Anonymous No.9463814]

>>9463797
u r thinking of a ring, friend

[Anonymous No.9463887]

>>9463508
yes, a semiring

[Anonymous No.9463900]

>>9463619
>A field is a group under both addition and multiplication.
Wrong.

[Anonymous No.9463915]

>>9463619

oh because 0 isn't invertible under multiplication
fuck I forgot that little point

[Anonymous No.9464039]

>>9463508
If there's no constraint on how the two operations have to interact, depending on the order of the group you can define many operations. If it needs to satisfy some distributive property between the two it depends.

[檢查人員:性感的腳步 No.9464065]

>>9463785
季節性數學 Because English is superior.

[Anonymous No.9464268]

>>9463508
Of course you can define different group structures on the same set. Simplest example would be [math]C_4[/math] and [math]V_4[/math]. But when you're defining the operation directly as a function on the set, it being closed under the operation is a given, and I think it is a mistake to list it among properties of groups. In general subgroups of a given group can be considered closed under an operation, and the group itself is trivially closed under its operation.

[Anonymous No.9464823]

>>9463785
you need a 会 in there..
我会说汉语

[Anonymous No.9465159]

>>9463814
>u r thinking of a ring, friend
Incorrect
You aren't thinking about the fact that 0 doesn't have a multiplicative inverse, friend.

[Anonymous No.9465168]

>>9463508
This thing oscilates back and froth on the screen, make it symmetric please :D

[Anonymous No.9465232]

Well for appropriately chosen n, R^n can have a lot of different non-isomorphic group structures. Eg: R^(2n^2) has the usual closedness under vector addition, but then you can isomorph it into the complex C^(n^2) which has a different addition, and further into M(n,R), the set of nxn matrices

[Anonymous No.9466439]

>>9465159
>doesn't have a multiplicative inverse
... do (You) therefore conclude the set of integers a is not a ring, friendo?

[〇中國經驗性中耕教師〇 No.9466481]

>>9464823
您的中国问候语:字沙拉。

[Anonymous No.9466539]

>>9466439
Are you actually this dumb? They're not saying you can't have a ring without multiplicative inverses, they're saying even a field isn't a group under its product because 0 doesn't have an inverse.