⚠ It’s A Trap ⚠

[Jon Awbrey]

Cf: ⚠ It’s A Trap ⚠
https://inquiryintoinquiry.com/2013/05/18/%e2%9a%a0-its-a-trap-%e2%9a%a0/

Re: Kenneth W. Regan • Graduate Student Traps
https://rjlipton.wpcomstaging.com/2013/05/15/graduate-student-traps

So it begins ...

The most common mathematical trap I run across has to do with
Triadic Relation Irreducibility, as noted and treated by the
polymath C.S. Peirce.

This trap lies in the mistaken belief that every
3-place (triadic or ternary) relation can be analyzed
purely in terms of 2-place (dyadic or binary) relations —
“purely” here meaning without resorting to any 3-place
relations in the process.

A notable thinker who not only fell but led many others
into this trap is none other than René Descartes, whose
problematic maxim I noted in the following post.

• Château Descartes
https://inquiryintoinquiry.com/2013/02/14/chateau-descartes/

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the
varieties of relational reduction that might be considered,
but usually we are talking about either one of two types
of reducibility.

• Compositional Reducibility.
All triadic relations are irreducible under relational composition,
since the composition of two dyadic relations is a dyadic relation,
by the definition of relational composition.

• Projective Reducibility.
Consider the projections of a triadic relation L ⊆ X × Y × Z
on the 3 coordinate planes X × Y, X × Z, Y × Z and ask whether
these dyadic relations uniquely determine L. If so, we say L is
“projectively reducible”, otherwise it is “projectively irreducible”.

Et Sic Deinceps …
=================

More Discussion of Relation Reduction
• https://oeis.org/wiki/Relation_reduction
• https://planetmath.org/RelationReduction

Previous Posts on Triadic Relation Irreducibility
• https://inquiryintoinquiry.com/2013/01/17/triadic-relation-irreducibility-1/
• https://inquiryintoinquiry.com/2013/01/18/triadic-relation-irreducibility-2/

Regards,

Jon

[Bernard Cohen]

The relational join operator, &, takes two binary relations to a triadic one, as defined thus:

- & - : (X « Y) x (X « Z) ® (X « (YxZ))

"x:X, y:Y, z:Z. (x,(y,z)) Î R&S Û xRz Ù ySz

Since the inverse may be defined similarly, any triadic relation T Í XxYxZ may be decomposed into 

the relational join R&S of two binary relations R Í XxY and S Í XxZ.

Patterns lively of the things rehearsed

On 16 Aug 2021, at 14:00, Jon Awbrey <jaw...@att.net> wrote:

Cf: ⚠ It’s A Trap ⚠

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[Jon Awbrey]

Hi Bernard,

The math symbols got so mutilated both via email and at the web interface
I can't tell which of many relational joins I know about is intended here.
Is there any sort of transcription to Ascii you could make or a link to a
web page for this particular join so I could think it over?

Regards,

Jon

[Cliff Joslyn]

On 8/16/2021 9:48 AM, Bernard Cohen wrote:

The relational join operator, &, takes two binary relations to a triadic one, as defined thus:

- & - : (X « Y) x (X « Z) ® (X « (YxZ))

"x:X, y:Y, z:Z. (x,(y,z)) Î R&S Û xRz Ù ySz

Since the inverse may be defined similarly, any triadic relation T Í XxYxZ may be decomposed into 

the relational join R&S of two binary relations R Í XxY and S Í XxZ.

Sorry for the typographical issues above, but it seems that by symmetry there are actually THREE decompositions for T \subseteq X x Y x Z:

R_1 \subseteq X x Y, S_1 \subseteq X x Z

R_2 \subseteq Y x X, S_1 \subseteq Y x Z

R_3 \subseteq Z x X, S_1 \subseteq Z X Y

correct?

To view this discussion on the web visit https://groups.google.com/d/msgid/cybcom/CANYEgbiRQLeDLY-vnQdu8tk6cK1iPApWcWgJ6fYaKjg3pin5uA%40mail.gmail.com.

-- 
O------------------------------------->
| Cliff Joslyn, Cybernetician at Large
V cajo...@gmail.com

[Cliff Joslyn]

LaTeX is effectively necessary in these situations. E.g. R \subseteq X
\times Y, x \in R, \vec{z} = (x,y), x \in X, y \in Y.

[Jon Awbrey]

Hi Bernie, Cliff ...

I keep a Unicode ref and frequently used symbols on this blog page:

• https://inquiryintoinquiry.com/toolbox/

What I was able to transcribe from the PDF Bernie sent was this.

<QUOTE ... sorta>

The relational join operator, & ,

takes two binary relations to a

triadic one, as defined (infix) thus:

- & - : (X ↔ Y) × (X ↔ Z) → (X ↔ (Y × Z))

∀x:X, y:Y, z:Z. (x,(y,z)) ∈ R&S ⇔ xRz ∧ ySz

Since the inverse may be defined similarly,

any triadic relation T ⊆ X×Y×Z may be decomposed

into the relational join R&S of two binary relations

R ⊆ X×Y and S ⊆ X×Z

</QUOTE>

I think this is an issue I know about and I'm in the
middle making up some Charts & Graphs to explain it.

Short answer for folks who know Lisp ... Cons !

But I wasn't sure about the meaning of the “↔” signs above,
or even whether they came across correctly in transmission ...
so that is still a question ...

Regards,

Jon