Relation Theory

[Jon Awbrey]

Cf: Survey of Relation Theory • 6
https://inquiryintoinquiry.com/2023/04/01/survey-of-relation-theory-6/

All,

In this Survey of blog and wiki posts on Relation Theory, relations are
viewed from the perspective of combinatorics, in other words, as a topic
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 is distinct from, though
closely related to, its study from the perspectives of abstract algebra on
the one hand and formal logic on the other.

Please follow the above link for the full set of resources.
A few basic articles are linked below.

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

Relational Concepts —
• Relation Construction ( https://oeis.org/wiki/Relation_construction )
• Relation Composition ( https://oeis.org/wiki/Relation_composition )
• Relation Reduction ( https://oeis.org/wiki/Relation_reduction )
• Relative Term ( https://oeis.org/wiki/Relative_term )
• Sign Relation ( https://oeis.org/wiki/Sign_relation )
• Triadic Relation ( https://oeis.org/wiki/Triadic_relation )
• Logic of Relatives ( https://oeis.org/wiki/Logic_of_relatives )
• Hypostatic Abstraction ( https://oeis.org/wiki/Hypostatic_abstraction )
• Continuous Predicate ( https://oeis.org/wiki/Continuous_predicate )

Illustrations —

Six Ways of Looking at a Triadic Relation ⌬ 1
• https://inquiryintoinquiry.com/2015/02/04/six-ways-of-looking-at-a-triadic-relation-%E2%8C%AC-1/

Peirce's 1870 “Logic of Relatives” —

Overview
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Preliminaries
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Regards,

Jon

[Ravi Sharma]

Jon

Is there also an attempt at integrating these relation concepts? like a meta-model of relations?

Thanks.

Ravi

(Dr. Ravi Sharma, Ph.D. USA)

NASA Apollo Achievement Award

Chair, Ontology Summit 2022

Particle and Space Physics

Senior Enterprise Architect

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[Jon Awbrey]

Cf: Relation Theory • Discussion 4
https://inquiryintoinquiry.com/2023/04/19/relation-theory-discussion-4/

Re: Survey of Relation Theory
https://inquiryintoinquiry.com/2023/04/01/survey-of-relation-theory-6/
Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/vlbILmZMsJ8
::: Ravi Sharma
https://groups.google.com/g/ontolog-forum/c/vlbILmZMsJ8/m/FWbax8syCAAJ

<QUOTE RS:>

Is there also an attempt at integrating these relation concepts?

Like a meta-model of relations?
</QUOTE>

Dear Ravi,

Thanks for the question. I believe I'd say yes to the
first part, since integrating diverse concepts is a big
part of what's called for here. At any rate, if we are
taking about the list of Relational Concepts at the top
of the above Survey Page then those are just so many
facets exhibited by the family of mathematical objects
known as relations.

Regards,

Jon

[Alex Shkotin]

Jon,

The structure which you describe as 

reminds me of my favorite structure "algebraic system", where we call every Xi the underlying set, carrier set or domain [1].

So you introduced a multi-carrier algebraic system with one relation in it.

As they call "relation" [2] the subset of the cartesian product which you call "graph". 

The algebraic system you describe is called relational as there are no operations in it.

Let me issue a hypothesis that any result in your Relations theory may be reformulated as a result in the theory of algebraic systems:-)

By the way, my experimental language YAFOLL is primarily to work with finite many-domain (aka sorts) algebraic systems [3].

One world, a lot of views:-)

Best regards,

Alex

[1] https://en.wikipedia.org/wiki/Algebraic_structure

[2] https://encyclopediaofmath.org/wiki/Algebraic_system

[3] https://www.researchgate.net/publication/307841408_Finite_Systems_Handling_Language_YAFOLL_message_1

вс, 16 апр. 2023 г. в 18:01, Jon Awbrey <jaw...@att.net>:

[Jon Awbrey]

Cf: Relation Theory • Discussion 5
https://inquiryintoinquiry.com/2023/04/22/relation-theory-discussion-5/

Re: Survey of Relation Theory
https://inquiryintoinquiry.com/2023/04/01/survey-of-relation-theory-6/
Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/vlbILmZMsJ8
::: Ravi Sharma
https://groups.google.com/g/ontolog-forum/c/vlbILmZMsJ8/m/FWbax8syCAAJ

<QUOTE RS:>
Is there also an attempt at integrating these relation concepts?

Like a meta‑model of relations?
</QUOTE>

Dear Ravi,

For the rest of your question, I haven't run across the concept
of a meta‑model before so I wasn't sure about that. If you get
a chance, maybe you can tell me more about that.

Many past discussions of models and their theories tell me a thicket of
failures to communicate is almost bound to arise at this point. One or
two pieces I've written in the past lay out the landscape a lot better
than I'm likely to do off the cuff, so I'll go look those up, but also
see if I can find a fresh perspective on the scene.

Regards,

Jon (https://mathstodon.xyz/@Inquiry)