When is equality not reflexive?

Question

In what fields, rings, groups, etc, is equality not symmetric? i.e.

$$x=y$$ but $$y\neq x$$

I have never heard of such a field/ring/group that exhibits this, but I know little about such things.

Edit: I'm asking this question because one of mathematics' main axioms is the axiom of reflexivity. I'm curious when it is ignored, if at all.

Answer 1

The equality symbol is always used to denote an equivalence relation.   Nominally the equivalence relation of the structure.

Equivalence relations are always reflexive, symmetric, and transitive.   It's what the term means.

It would be fairly obtuse to use the symbol for any other purpose.

Answer 2

In short (TLDR), equality is always symmetric because of how it is defined. (It is defined with the logical biconditional operator $\iff$, which is itself symmetric.)

Long answer:

There are two ways I have seen equality defined. The first way is Predicate Logic with Equality, and the second way is Predicate Logic without Equality. Both ways use $\iff$, which is symmetric (that is, $A \iff B$ is equivalent to $B \iff A$).

Predicate Logic with Equality

defines equality as a sort-of “axiom,” that equal objects are identical. Here, identical means that if $x$ and $y$ are identical, any property that holds for $x$ also holds for $y$ and vice-versa.

$$ (x=y) \implies (P(x) \iff P(y)) \qquad (1) $$

This principle is known as the Indiscernibility of Identicals: objects that are identical are also indiscernible, a pretty straightforward concept.

Clearly, equality is symmetric because $\iff$ is also symmetric.

Two special cases of this axiom are in second-order logic where the predicate $P(z)$ could stand for “some set $T$ is in $z$” or “$z$ is in some set $S$,” etc. The predicate is just a placeholder for any statement. The two lines below are specific examples.

\begin{align} (x=y) &\implies (\forall T)(T\in x \iff T\in y) \qquad (2) \\ (x=y) &\implies (\forall S)(x\in S \iff y\in S) \qquad (3) \\ &\text{etc.} \end{align}

Also in Second-Order Logic derived from Predicate Logic with Equality, there is an Axiom of Extensionality. This states the converse of line (2) above, and is also known as the Identicality of Indiscernibles: objects that are indiscernible are also identical.

$$ (\forall T)(T\in x \iff T\in y) \implies (x=y) \qquad (4)$$

In other words, sets that have exactly the same elements are equal.

(Let it be noted here that this principle is not as straightforward as it sounds and should not be taken for granted. For example, in some topologies, points that are topologically indistinguishable are not always equal. Thus in second-order logic this axiom is very beneficial.)

Predicate Logic without Equality

The second way I have seen is where Predicate Logic itself is written without any concept of equality, and where authors will wait until Second-Order Logic to rigorously define equality, which doesn't rely on the intuition of “identicalness” or on predicates. In fact, it actually uses a merger of both lines (2) and (4) above. Thus in this system the very definition of $x=y$ is:

$$ (x=y) \iff (\forall T)(T\in x \iff T\in y) \qquad (5) $$

We can also see here that equality is symmetric, as before, because $\iff$ is symmetric.

This system then goes on to define the Axiom of Intensionality (so appropriately named):

$$ (x=y) \iff (\forall S)(x\in S \iff y\in S) \qquad (6)$$

(sets that are members of the same supersets are equal), which I think is quite fascinating because we don't typically think about equal sets in this way.