Thread: A mathematics system without identity rule?

Hi

What happen if i drop the identity rule in the definition of a group?

What happen if a mathematical system does not have identity rule?

Are there any mathematical system that doesn't have identity rule?

That's too vague to say anything of substance to in reply.
However, in general, the identity represents the "change nothing" operation/element/whatever, and so is almost always found in math structures.

The one place I can recall where the identity was notably missing was in Ring Theory, where the *multiplicative* identity was sometimes excluded. I saw them called "rngs" rather than "rings" (get it? no "i" - hahaha, those witty mathematicians), but it's common to hear "ring with identity" to remove any uncertainty. However, even for rngs, there was always the identity of the additive operation (rng, + , 0).

Originally Posted by erich22

What happen if i drop the identity rule in the definition of a group?

What happen if a mathematical system does not have identity rule?

Are there any mathematical system that doesn't have identity rule?

I absolutely agree with the statement:
"That's too vague to say anything of substance to in reply."

Here are comments.
The concept of identity is associated with an operation.
There are operations without identities.
Do you mean to say that all mathematical system must have operation(s)?

Thank you very much, john.
I've just read the requirement of a vector space, i just wonder if the identity rule is really needed? what about the requirement in a group, i just wonder, thanks for the answer. Hmmmm, so there is a system like rngs () that doesn't have identity rule.
Ok, thank you so much, i want to investigate the rngs.

Thank you so much Plato.
Ok, I get it, so identity must be associated with an operation.
Could you tell me any mathematical system that doesn't have identity rule except the rngs mentioned by john. I want to study this topic a little further through the examples.
Your post makes me wondering, are there any mathematical system that doesn't have any operation?

A semigroup is a group without identity or inverse, i.e. only with associativity.

Thanks emakarov, so what i need to see is semigroup. Ok.