What axioms does equality have?

Question

What axioms does equality have, I found different answers.

If $a=b$ $a,b$ are mathematical objects. What can we claim about it?

Can we calim that if $o$ is an operation then $o(a)=o(b)$?

Answer 1

“equality is a relationship…asserting…that the expressions represent the same mathematical object” - from Wikipedia. So, looking from the perspective of the function $o$, it doesn’t even know that the $a$ and $b$ are different items, as equality claims them to be the same. So $o(a)=o(b)$ is true.

This meaning of equality stems from the fact that $a$ and $b$ are called equal only if $a$ and $b$ are just labels for the same underlying object. A mathematical object can be anything; as long as you take one and call it $a$ in one context and $b$ in another, $a=b$ holds true.

However, there might be restrictions on the equality that we use, extending its definition, so that equality is not all-encompassing any more. Instead we can restrict it to be defined such that $a$ and $b$ are called equal under $K$ if a particular property $K$ of $a$ and $b$ are equal.

Take the example of equality under similarity (for triangles). A large equilateral triangle drawn on a rough sheet and another equilateral triangle drawn on my nail are equal under similarity, but not under congruence.