Talk:Incidence (geometry)

PG(2,F)

Read definition:

Let V be the three dimensional vector space defined over the field F. The projective plane P(V) = PG(2, F) consists of the one dimensional vector subspaces of V called points and the two dimensional vector subspaces of V called lines...

All vector subspaces in this definition must have common point. In other case parallel (in V) planes will not meet, and corresponding projective lines will not meet. So, it will not be projective plane. Jumpow (talk) 07:39, 22 August 2017 (UTC)

All the vector subspaces have a common vector (the zero vector), but this is not a point, since, by itself, it is a zero-dimensional subspace. In a three-dimensional space, two 2-dimensional subspaces must meet in a 1-dimensional subspace. This follows from dim(S + T) = dim(S) + dim(T) - dim(S∩T), for any subspaces S and T. --Bill Cherowitzo (talk) 18:02, 22 August 2017 (UTC)

You are wrong. All vector subspaces in this definition must have common point, not pairwise. Jumpow (talk) 08:05, 23 August 2017 (UTC)

I understood... Sorry. Here keyword vector subspaces, not just subspaces. Jumpow (talk) 08:10, 23 August 2017 (UTC)

Symmetric language

The word incident provides symmetric terminology, but the incidence relation is not a symmetric relation unless it is self-dual, such as in the Fano plane. Reference to incidence structure#Dual structure shows the swapping of points and blocks, and the possible isomorphism resulting in symmetry. For a relation to be symmetric its logical matrix must equal its transpose, a rare property of heterogeneous relations.

When using a phrase such as "In the literature this is referred to as ...", the project expects an author, title, and page reference. Pronouncements without backup are suspect. — Rgdboer (talk) 22:15, 30 July 2018 (UTC)

Perhaps I was being a little too concise in not expanding on the idea that the use of "symmetry" was more of a suggestive terminology than a precisely defined property of relations. I did not want to belabor the point, but I felt that it was necessary to point out that in the incidence geometry literature one was not going to find the term "heterogeneous relation" as this is a relatively new term introduced by computer scientists. Geometric language predates the set-theoretic language of relations and the fit between them had a couple of rough spots. The expressions "a point P lies on a line $\ell$" and "the line $\ell$ passes through the point P" are just two ways to express the same incidence relation. If this relation is denoted by the symbol "I", then it is clear that we should write "P I $\ell$ " if and only if "$\ell$ I P " to symbolize these statements and this certainly looks like a symmetric relation. There have been attempts, most notably G. Pickert, Projecktive Ebenen, p.2, to make this an actual symmetric relation by defining the incidence relation on the set ${\mathcal {P}}\cup {\mathcal {L}}$, where ${\mathcal {P}}$ is the point set and ${\mathcal {L}}$ is the line set. Other authors have tried different means to get around this sticky intersection of terminologies. I find most of these attempts to be kludges and I am happy to see that someone is attempting to straighten out and unify the relation terminology, but not all such attempts in the past have caught on (for instance, there is nothing symmetric about a symmetric block design and many attempts have been made to get people to change the terminology, to no avail) and we do a disservice to our readers if we adopt the current reforms without pointing out what is actually in the literature. Remember, "Pronouncements without backup are suspect."! --Bill Cherowitzo (talk) 20:21, 31 July 2018 (UTC)

Just a reminder: Binary relation refers to ordered pair and (a,b) ≠ (b,a). — Rgdboer (talk) 22:42, 4 August 2018 (UTC)

Which just goes to prove my point. Binary relation between sets is a common locution used in this area. The references cited in this article attest to this fact. Moorhouse, p. 3 and Beth, Jungnickel and Lenz, p. 15. --Bill Cherowitzo (talk) 19:35, 5 August 2018 (UTC)

Heterorelativ: 1895 by Ernst Schroder Algebra der Logik, Band III page 12. — Rgdboer (talk) 21:51, 16 August 2018 (UTC)