Composition iterates, Cauchy, translation, and Sincov inclusions

Abstract

Abstract Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕn of a relation ϕ on X, defined by ϕ 0 = Δ x ,     ϕ n = ϕ ∘ ϕ n - 1  if n ∈ 𝕅 ,     and      ϕ ∞ = ∪ n = 0 ∞ ϕ n . {\varphi ^0} = {\Delta _x},\,\,{\varphi ^n} = \varphi \circ {\varphi ^{n - 1}}{\rm{ if n}} \in \mathbb{N,}\,\,{\rm{and }}\,\,{\varphi ^\infty } = \bigcup\limits_{n = 0}^\infty {{\varphi ^n}} . In particular, by using the relational inclusion ϕn ◦ϕm ⊆ ϕn+m with n, m ∈ 𝕅 ¯ 0 \mathbb{\bar {N}_0}} , we show that the function α, defined by α ( n ) = ϕ n       for  n ∈ 𝕅 ¯ 0 , \alpha \left( n \right) = {\varphi ^{\rm{n}}}\,\,\,{\rm{for n}} \in {{\rm\mathbb{\bar N}}_{\rm{0}}}, satisfies the Cauchy problem α ( n ) ∘ α ( m ) ⊆ α ( n + m ) ,       α ( 0 ) = Δ x . \alpha \left( n \right) \circ \alpha \left( {\rm{m}} \right) \subseteq \alpha \left( {{\rm{n}} + {\rm{m}}} \right),\,\,\,\alpha \left( 0 \right) = {\Delta _{\rm{x}}}. Moreover, the function f, defined by f ( n , A ) = α ( n ) [ A ]       for   n ∈ 𝕅 ¯ 0     and   A ⊆ X , {\rm{f}}\left( {{\rm{n}},{\rm{A}}} \right) = \alpha \left( {\rm{n}} \right)\left[ {\rm{A}} \right]\,\,\,{\rm{for}}\,{\rm{n}} \in {{\rm\mathbb{\bar {N}}}_{\rm{0}}}\,\,{\rm{and}}\,{\rm{A}} \subseteq {\rm{X,}} satisfies the translation problem f ( n , f ( m , A) ) ⊆ f ( n + m , A ) ,       f ( 0 , A ) = A . {\rm{f}}\left( {{\rm{n}},f(m,{\rm{A)}}} \right) \subseteq {\rm{f}}\left( {{\rm{n}} + {\rm{m,A}}} \right),\,\,\,{\rm{f}}\left( {0,{\rm{A}}} \right) = {\rm{A}}{\rm{.}} Furthermore, the function F, defined by F ( A , B ) = { n ∈ 𝕅 ¯ 0 :     A ⊆ f ( n , B ) }     for     A , B ⊆ X , {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) = \left\{ {{\rm{n}} \in {{{\rm\mathbb{\bar {N}}}}_{\rm{0}}}:\,\,{\rm{A}} \subseteq {\rm{f}}\left( {{\rm{n}},{\rm{B}}} \right)} \right\}\,\,{\rm{for}}\,\,{\rm{A,B}} \subseteq {\rm{X,}} satisfies the Sincov problem F ( A , B ) + F ( B , C ) ⊆ F ( A , C ) ,         0 ∈ F ( A , A ) . {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) + {\rm{F}}\left( {{\rm{B}},{\rm{C}}} \right) \subseteq {\rm{F}}\left( {{\rm{A,C}}} \right),\,\,\,\,0 \in {\rm{F}}\left( {{\rm{A}},{\rm{A}}} \right). Motivated by the above observations, we investigate a function F on the product set X2 to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that F ( x , y ) + F ( y , z ) ⊆ F ( x , z ) {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{F}}\left( {{\rm{y}},{\rm{z}}} \right) \subseteq {\rm{F}}\left( {{\rm{x}},{\rm{z}}} \right) for all x, y, z ∈ X. For this, we introduce the convenient notations R ( x , y ) = F ( y , x )       and     S ( x , y ) = F ( x , y ) + R ( x , y ) , {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{y}},{\rm{x}}} \right)\,\,\,{\rm{and}}\,\,{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right), and Φ ( x ) = F ( x , x )     and     Ψ ( x ) ∪ y ∈ X S ( x , y ) . \Phi \left( {\rm{x}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{x}}} \right)\,\,{\rm{and}}\,\,\Psi \left( {\rm{x}} \right)\bigcup\limits_{{\rm{y}} \in {\rm{X}}} {{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right).} Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric, or single-valued.