[CG] Re: Relation Theory

Cf: Relation Theory • 3 https://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 ========= • 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