Deleting duplicate equations from a list of equations

Here is a list of algebraicly identical equations, and I like to delete all the duplicate equations from this list and get only two equations. I tried the following:

Clear[ss];
listEqs = {
x + y == x^2 + z,
y + z == x^2 + z,
x + y == z + x^2,
y + z == z + x^2,
x^2 + z == x + y,
z + x^2 == x + y,
x^2 + z == y + z,
z + x^2 == y + x
};

DeleteDuplicates[listEqs]


I got:

{
x + y == x^2 + z,
y + z == x^2 + z,
x^2 + z == x + y,
x^2 + z == y + z
}


Can someone tell me how to reduce the list to only two equations?

EDIT 1

I noticed a misunderstanding of my question. Here is a second list:

listEqs = {
y + z == x^2 + z,
a + b == k + g,
x + y == x^2 + z,
b + a == g + k,
x + y == z + x^2,
a + b == g + k,
y + z == z + x^2,
x^2 + z == x + y,
z + x^2 == x + y,
x^2 + z == y + z,
z + x^2 == y + x};


The code:

DeleteDuplicatesBy[listEqs, MemberQ@Level[listEqs, -1]]


picks the first element of the list, which is not what I asked. I want the following list to be the output of the code:

{
y + z == x^2 + z,
a + b == k + g,
x + y == x^2 + z
}


Basically, I just want to pick the non-identical equations from the list. The desired equations can be placed in the middle, end or beginning of the list. The position of the equations are not the issue.

• listEqs // Simplify // Union – Bob Hanlon Oct 20 '18 at 23:27
• @BobHanlon: Your proposal is not quite what I want since it creates new equations by cancelling the common terms on both sides of the equations, a result of Simplify command. Thanks for your suggestion. – Tugrul Temel Oct 20 '18 at 23:59

Maybe

DeleteDuplicatesBy[listEqs, Reduce] (* or *)
DeleteDuplicatesBy[listEqs, Sort]


{x + y == x^2 + z, y + z == x^2 + z}

For the longer input list:

DeleteDuplicatesBy[listEqs2, Reduce]


{y + z == x^2 + z, a + b == g + k, x + y == x^2 + z}

• It is a typo. Sorry for confusion, but your answer is the right one and it works for me. Thank you very much. – Tugrul Temel Oct 20 '18 at 21:05