Relation Theory

[Jon Awbrey]

Cf: Relation Theory • 1
https://inquiryintoinquiry.com/2021/10/27/relation-theory-1/

All,

Here's an introduction to Relation Theory geared to applications and
taking a moderately general view at least as far as finite numbers
of relational domains are concerned (k-adic or k-ary relations).

Relation Theory ( https://oeis.org/wiki/Relation_theory )

This article treats relations from the perspective of combinatorics,
in other words, as a subject matter in discrete mathematics, with
special attention to finite structures and concrete set-theoretic
constructions, many of which arise quite naturally in applications.
This approach to relation theory, or the theory of relations, is
distinguished from, though closely related to, its study from the
perspectives of abstract algebra on the one hand and formal logic
on the other.

Resources
=========

• Relation Theory ( https://oeis.org/wiki/Relation_theory )
• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )

• Survey of Relation Theory
( https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/ )

• Peirce's 1870 Logic Of Relatives
( https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview )

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • 2
https://inquiryintoinquiry.com/2021/10/28/relation-theory-2/

All,

I am putting a new edition of my Relation Theory article on my blog
and transcribing a plaintext copy for whatever discussion may arise,
but please see the above-linked blog post for better-formatted copy.

Preliminaries
=============
https://oeis.org/wiki/Relation_theory#Preliminaries

Two definitions of the relation concept are common in the literature.
Although it is usually clear in context which definition is being used
at a given time, it tends to become less clear as contexts collide, or
as discussion moves from one context to another.

The same sort of ambiguity arose in the development of the function concept
and it may save a measure of effort to follow the pattern of resolution that
worked itself out there.

When we speak of a function f : X → Y we are thinking of
a mathematical object whose articulation requires three
pieces of data, specifying the set X, the set Y, and
a particular subset of their cartesian product X × Y.
So far so good.

Let us write f = (obj₁f, obj₂f, obj₁₂f) to express what has been said so far.

When it comes to parsing the notation “f : X → Y”, everyone takes the part
“X → Y” as indicating the type of the function, in effect defining type(f)
as the pair (obj₁f, obj₂f), but “f” is used equivocally to denote both the
triple (obj₁f, obj₂f, obj₁₂f) and the subset obj₁₂f forming one part of it.

One way to resolve the ambiguity is to formalize a distinction between the
function f = (obj₁f, obj₂f, obj₁₂f) and its “graph”, defining graph(f) = obj₁₂f.

Another tactic treats the whole notation “f : X → Y” as a name for the triple,
letting “f” denote graph(f).

In categorical and computational contexts, at least initially, the type is
regarded as an essential attribute or integral part of the function itself.
In other contexts we may wish to use a more abstract concept of function,
treating a function as a mathematical object capable of being viewed under
many different types.

Following the pattern of the functional case, let the notation “L ⊆ X × Y”
bring to mind a mathematical object which is specified by three pieces of data,
the set X, the set Y, and a particular subset of their cartesian product X × Y.
As before we have two choices, either let L = (X, Y, graph(L)) or let “L” denote
graph(L) and choose another name for the triple.

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • 3
http://inquiryintoinquiry.com/2021/10/29/relation-theory-3/

All,

It is convenient to begin with the definition of a k-place relation,
where k is a positive integer.

Definition. A k-place relation L ⊆ X₁ × … × Xₖ over the
nonempty sets X₁, …, Xₖ is a (k+1)-tuple (X₁, …, Xₖ, L)
where L is a subset of the cartesian product X₁ × … × Xₖ.

Several items of terminology are useful in discussing relations.

• The sets X₁, …, Xₖ are called the “domains” of the
relation L ⊆ X₁ × … × Xₖ, with Xₘ being the m-th domain.

• If all the Xₘ are the same set X then L ⊆ X₁ × … × Xₖ
is more simply described as a k-place relation over X.

• The set L is called the “graph” of the relation
L ⊆ X₁ × … × Xₖ, on analogy with the graph of
a function.

• If the sequence of sets X₁, …, Xₖ is constant throughout a given
discussion or is otherwise determinate in context then the relation
L ⊆ X₁ × … × Xₖ is determined by its graph L, making it acceptable to
denote the relation by referring to its graph.

• Other synonyms for the adjective “k-place” are “k-adic” and “k-ary”,
all of which leads to the integer k being called the “dimension”,
“adicity”, or “arity” of the relation L.

Resources
=========

• Survey of Relation Theory
( https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/ )

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • 4
http://inquiryintoinquiry.com/2021/10/30/relation-theory-4/

All,

The next few definitions of “local incidence properties” of relations
are given at a moderate level of generality in order to show how they
apply to k-place relations. In the sequel we'll see what light they
throw on a number of more familiar 2-place relations and functions.

A “local incidence property” of a relation L is a property which
depends in turn on the properties of special subsets of L known
as its “local flags”. The local flags of a relation are defined
in the following way.

Let L be a k-place relation L ⊆ X₁ × … × Xₖ.

Pick a relational domain Xₘ and a point x in Xₘ.

The “flag of L with x at m”, written Lₓ@ₘ and also known as the
“x@m-flag of L”, is a subset of L with the following definition.

• Lₓ@ₘ = {(x₁, …, xₘ …, xₖ) ∈ L : xₘ = x}.

Any property C of the local flag Lₓ@ₘ is said to be a
“local incidence property of L with respect to the locus x @ m”.

A k-adic relation L ⊆ X₁ × … × Xₖ is said to be “C-regular at m”
if and only if every flag of L with x at m has the property C,
where x is taken to vary over the “theme” of the fixed domain Xₘ.

Expressed in symbols, L is C-regular at m
if and only if C(Lₓ@ₘ) is true for all x in Xₘ.

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • 5
https://inquiryintoinquiry.com/2021/10/31/relation-theory-5/

All,

Two further classes of incidence properties will prove to be of great utility.

Regional Incidence Properties
=============================
https://oeis.org/wiki/Relation_theory#Regional_incidence_properties

The definition of a local flag can be broadened from
a point to a subset of a relational domain, arriving at
the definition of a “regional flag” in the following way.

Let L be a k-place relation L ⊆ X₁ × … × Xₖ.

Choose a relational domain Xₘ and one of its subsets S ⊆ Xₘ.

Then L_{S @ m} is a subset of L called the flag of L with S at m,
or the (S @ m)-flag of L, a mathematical object with the following
definition.

• L_{S @ m} = {(x₁, …, xₘ …, xₖ) ∈ L : xₘ ∈ S}.

Numerical Incidence Properties
==============================
https://oeis.org/wiki/Relation_theory#Numerical_incidence_properties

A “numerical incidence property” of a relation
is a local incidence property predicated on
the cardinalities of its local flags.

For example, L is said to be “c-regular at m” if and only if
the cardinality of the local flag Lₓ@ₘ is c for all x in Xₘ —
to write it in symbols, if and only if |Lₓ@ₘ| = c for all x ∈ Xₘ.

In a similar fashion, one may define the numerical incidence properties,
(<c)-regular at m, (>c)-regular at m, and so on. For ease of reference,
a few definitions are recorded below.

Numerical Incidence Properties
https://inquiryintoinquiry.files.wordpress.com/2021/10/numerical-incidence-properties-alt.png

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • Discussion 2
https://inquiryintoinquiry.com/2021/11/01/relation-theory-discussion-2/

Re: Relation Theory • 4
https://inquiryintoinquiry.com/2021/10/30/relation-theory-4/
Re: FB | Charles S. Peirce Society
https://www.facebook.com/groups/peircesociety/posts/2457168684419109/
::: Joseph Harry
https://www.facebook.com/groups/peircesociety/posts/2457168684419109?comment_id=2457724677696843

<QUOTE JH:>
These are iconic representations dealing with logical symbolic relations,
and so of course are semiotic in Peirce's sense, since logic is semiotic.
But couldn't a logician do all of this meticulous formalization and
understand all of the discrete logical consequences of it without
having any inkling of semiotics or of Peirce?

Dear Joseph,

As I noted at the top of the article and blog series —
https://inquiryintoinquiry.com/2021/10/27/relation-theory-1/

“This article treats relations from the perspective of combinatorics,

in other words, as a subject matter in discrete mathematics, with
special attention to finite structures and concrete set-theoretic
constructions, many of which arise quite naturally in applications.
This approach to relation theory, or the theory of relations, is
distinguished from, though closely related to, its study from the
perspectives of abstract algebra on the one hand and formal logic

on the other.”

Of course one can always pull a logical formalism out of thin air, with no inkling
of its historical sources, and proceed in a blithely syntactic and deductive fashion.
But if we hew more closely to applications, original or potential, and even regard
logic and math as springing from practice, we must take care for the semantic and
pragmatic grounds of their use. From that perspective, models come first, well
before the deductive theories whose consistency they establish.

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • Discussion 3
https://inquiryintoinquiry.com/2021/11/03/relation-theory-discussion-3/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/relation_theory/86544796
::: James Bowery ( https://groups.io/g/lawsofform/message/1116

<QUOTE JB:>
Thanks for that very rigorous definition of “relation theory”.

Its “trick” of including the name of the k-relation in a (k+1)-relation’s
tuples reminds me Etter’s paper “Three-Place Identity” which was the result
of some of our work at HP on dealing with identity (starting with the very
practical need to identify individuals/corporations, etc. for the purpose
of permitting meta-data that attributed assertions of fact to certain
identities aka “provenance” of data).

The result of that effort threatens to up-end set theory itself
and was to be fully fleshed out in “Membership and Identity” […]

We were able to get a preliminary review of Three-Place Identity by a close
associate of Ray Smullyan. It came back with a positive verdict. I believe
I may still have that letter somewhere in my archives.
</QUOTE>

Dear James,

The article on Relation Theory ( https://oeis.org/wiki/Relation_theory )
represents my attempt to bridge the two cultures of weak typing and
strong typing approaches to functions and relations. Weak typing was
taught in those halcyon Halmos days when functions and relations were
nothing but subsets of cartesian products. Strong typing came to the
fore with category theory, its arrows from source to target domains,
and the need for closely watched domains in computer science.

Peirce recognizes a fundamental triadic relation he calls “teridentity”
where three variables a, b, c denote the same object, represented in
his logical graphs as a node of degree three, and at first I thought
you might be talking about that.

But I see x(y = z) read as “x regards y as the same as z” is more like
the expressions I use to discuss “equivalence relations from a particular
point of view”, following one of Peirce’s more radical innovations from his
1870 “Logic of Relatives”.

• C.S. Peirce • On the Doctrine of Individuals
https://inquiryintoinquiry.com/2015/02/23/mathematical-demonstration-the-doctrine-of-individuals-2/

Using square brackets in the form [a]_e for the equivalence class of
an element a in an equivalence relation e we can express the above idea
in one of the following forms.

• [y = z]_x

• [y]_x = [z]_x

• y =_x z

I wrote this up in general somewhere but there’s a fair enough illustration of
the main idea in the following application to “semiotic equivalence relations”.

• Semiotic Equivalence Relations
https://oeis.org/wiki/Sign_relation#Semiotic_equivalence_relations

The rest of your remarks bring up a wealth of associations for me,
as seeing the triadic unity in the multiplicity of dyadic appearances
is a lot of what the Peircean perspective is all about. I’ll have to
dig up a few old links to fill that out …

Regards,

Jon

[Jon Awbrey]

Cf: Relation Theory • 6
https://inquiryintoinquiry.com/2021/11/04/relation-theory-6/

All,

Let's take a look at a few old friends from the class of 2-place relations
as they appear against the backdrop of our current view on relation theory.

Species of Dyadic Relations
===========================
https://oeis.org/wiki/Relation_theory#Species_of_dyadic_relations

Returning to 2-adic relations, it is useful to describe
several familiar classes of objects in terms of their
local and numerical incidence properties.

Let L ⊆ S × T be an arbitrary 2-adic relation.
The following properties of L can be defined.

Display 1. Dyadic Relations • Total • Tubular
https://inquiryintoinquiry.files.wordpress.com/2021/11/dyadic-relations-e280a2-total-e280a2-tubular.png

If L ⊆ S × T is tubular at S then L is called a “partial function” or
a “prefunction” from S to T. This is sometimes indicated by giving L
an alternate name, for example, “p”, and writing L = p : S ⇀ T.

Just by way of formalizing the definition:

• L = p : S ⇀ T if and only if L is tubular at S.

If L is a prefunction p : S ⇀ T which happens to be total at S,
then L is called a “function” from S to T, indicated by writing
L = f : S → T. To say a relation L ⊆ S × T is totally tubular at S
is to say it is 1-regular at S. Thus, we may formalize the following
definition.

• L = f : S → T if and only if L is 1-regular a S.

In the case of a function f : S → T,
we have the following additional definitions.

Display 2. Dyadic Relations • Surjective, Injective, Bijective
https://inquiryintoinquiry.files.wordpress.com/2021/11/dyadic-relations-e280a2-surjective-injective-bijective.png

Resources
=========

https://inquiryintoinquiry.com/2020/05/15/survey-of-relation-theory-4/

Regards,

Jon