Composition iterates, Cauchy, translation, and Sincov inclusions

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 ϕ0=Δx, ϕn=ϕ∘ϕn-1 if n∈𝕅, and ϕ∞=∪n=0∞ϕn.{\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◦ϕ^m⊆ ϕ^n+m with n, m ∈ 𝕅¯0\mathbb{\bar {N}_0}}, we show that the function α, defined by α(n)=ϕn 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)=Δx.\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∈XS(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.