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).