I am trying to write a first order logic formula for the following: Let L = [C,=] be a vocabulary with C, a ternary relation symbol. Let K be the class of finite L-structures $\mathcal A$ in which A = [n] = [1,...n] and for each pair(a,b) $\in$ A, there is exactly one c ∈ A such that (a,b,c) $\in$ $C^{\mathcal A}$ (So C is like 2 dimensional matrix with indices and element all drawn from the same set A, where C(i,j,z) means the entry in $C_{i,j}$ is z.)
I want to write a formula with one free variable x, suich that in any structure $A$ $\in$ K, the formula defines the set of values that appear on the main diagonal of the matrix $C^{\mathcal A}$. I am not really sure how I could write this in first order logic. Any help would be greatly appreciated!
Since $C(x,y,z)$ is true if $z$ is in the $(x,y)$th position of the matrix, and $z$ is on the diagonal whenever $z$ is the $(x,x)$th position for some $x$, we can see $z$ is on the diagonal if and only if $C(x,x,z)$ is true for some $x$.
Can you find a way to express this?
I hope this helps ^_^