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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?

120MinuteMan
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1 Answers1

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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 $P \times P$.

arkeet
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  • From reading your answer, I understand that I misunderstood the question. +1 – Axoren Sep 28 '16 at 02:18
  • Crystal clear now, thanks! – 120MinuteMan Sep 28 '16 at 02:20
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    BTW, I'm inclined to call the diagonal the "identity relation", as it acts as an identity for relation composition. But that would be confusing, given the author's terminology. – arkeet Sep 28 '16 at 02:24
  • The terminology does not seem to be firmly established. I would think that the diagonal is same as the identity (or equality) relation, as in https://proofwiki.org/wiki/Definition:Diagonal_Relation . I would call $P\times P$ total relation, but this conflicts with the definition of a total order, others suggest to use the term universal relations, see https://en.wikipedia.org/wiki/Total_relation and https://en.wikipedia.org/wiki/Binary_relation , question and answers https://math.stackexchange.com/q/461233 and https://math.stackexchange.com/q/461235 and https://math.stackexchange.com/q/461241 – Mirko Feb 26 '18 at 05:51
  • I do not know why the author chose the term identity relation for $P\times P$, but I do see one possible reason. In some (older) books set intersection is denoted and thought of like multiplication. More so in lattice theory, where intersection goes with meet and with Boolean multiplication. The relation $P\times P$ is indeed the identity with respect to this kind of multiplication, that is, for any relation $R$ we have $R=R\cap(P\times P)=R\wedge(P\times P)=R\cdot(P\times P)$. – Mirko Feb 26 '18 at 06:03