[CG] Re: Relation Theory

Cf: Relation Theory • 4 https://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. 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/2021/11/08/survey-of-relation-theory-5/ Peirce's 1870 “Logic of Relatives” https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-p… Regards, Jon