I am reading through Skornjakov's Elements of Lattice Theory (1977), and am having trouble understanding what is meant by a diagonal relation. I am also unclear about how the diagonal differs from the identity.
The book defines a diagonal and identity relation as follows:
The relation consisting of all pairs $(a,a)$, where $a \in P$, is called the diagonal and denoted by $\Delta$. The relation coinciding with the entire set $P \times P$ is called the identity.
I am trying to think through this with two basic examples. Take the set $A = \{1,2,3\}$ and $B = \{5\}$. What are the diagonal and identity of $A \times A$? What are the diagonal and identity of $B \times B$?
My sense of this is that the identity of $A \times A$ is $\{(1,1),(2,2),(3,3)\}$ and the identity of $B \times B$ is $\{5\}$. I am not sure what the diagonals are?
The identity of $P \times P$ is just as it says; it is the entire set $P \times P$. (I have no idea why it's called that.) So it has $|P|^2$ elements.
The diagonal of $P \times P$ is the set $\{(a,a) : a \in P\}$, so it has $|P|$ elements. Another way it can be written is $\{(a,b) : a,b \in P, a = b\}$. The reason it is called the diagonal is that, for instance when $P = \mathbb{R}$ is the real line, the diagonal is the graph of $y = x$ in $\mathbb{R} \times \mathbb{R}$ -- it's a diagonal line.
Note that they are both subsets of $P \times P$.