<abstract xmlns="http://www.w3.org/1999/xhtml">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
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>ϕ</mi></mrow><mn>0</mn></msup><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mi>x</mi></msub><mo>,</mo><mi> </mi><mi> </mi><msup><mrow><mi>ϕ</mi></mrow><mi>n</mi></msup><mo>=</mo><mi>ϕ</mi><mo>∘</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mtext> if n</mtext><mo>∈</mo><mtext>𝕅</mtext><mo>,</mo><mi> </mi><mi> </mi><mtext>and </mtext><mi> </mi><mi> </mi><msup><mrow><mi>ϕ</mi></mrow><mo>∞</mo></msup><mo>=</mo><munderover><mo>∪</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></munderover><mrow><msup><mrow><mi>ϕ</mi></mrow><mi>n</mi></msup></mrow><mo>.</mo></mrow></math><tex-math>{\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}} .</tex-math></alternatives></disp-formula>In particular, by using the relational inclusion ϕn<italic>◦</italic>ϕm<italic>⊆</italic> ϕn+m with n, m ∈ <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mover><mi>𝕅</mi><mo>¯</mo></mover></mrow><mn>0</mn></msub></mrow></math><tex-math>\mathbb{\bar {N}_0}}</tex-math></alternatives></inline-formula>, we show that the function α, defined by
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>ϕ</mi></mrow><mtext>n</mtext></msup><mi> </mi><mi> </mi><mi> </mi><mtext>for  n</mtext><mo>∈</mo><msub><mrow><mover><mi>𝕅</mi><mo>¯</mo></mover></mrow><mtext>0</mtext></msub><mo>,</mo></mrow></math><tex-math>\alpha \left( n \right) = {\varphi ^{\rm{n}}}\,\,\,{\rm{for n}} \in {{\rm\mathbb{\bar N}}_{\rm{0}}},</tex-math></alternatives></disp-formula>
satisfies the Cauchy problem
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_004.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>∘</mo><mi>α</mi><mrow><mo>(</mo><mtext>m</mtext><mo>)</mo></mrow><mo>⊆</mo><mi>α</mi><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>+</mo><mtext>m</mtext></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>α</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mtext>x</mtext></msub><mo>.</mo></mrow></math><tex-math>\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}}}.</tex-math></alternatives></disp-formula>Moreover, the function f, defined by
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_005.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>,</mo><mtext>A</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mrow><mo>(</mo><mtext>n</mtext><mo>)</mo></mrow><mrow><mo>[</mo><mtext>A</mtext><mo>]</mo></mrow><mi> </mi><mi> </mi><mi> </mi><mtext>for</mtext><mi> </mi><mtext>n</mtext><mo>∈</mo><msub><mrow><mover><mtext>𝕅</mtext><mo>¯</mo></mover></mrow><mtext>0</mtext></msub><mi> </mi><mi> </mi><mtext>and</mtext><mi> </mi><mtext>A</mtext><mo>⊆</mo><mtext>X</mtext><mo>,</mo></mrow></math><tex-math>{\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,}}</tex-math></alternatives></disp-formula>
satisfies the translation problem
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>,</mo><mi>f</mi><mo>(</mo><mi>m</mi><mo>,</mo><mtext>A)</mtext></mrow><mo>)</mo></mrow><mo>⊆</mo><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>+</mo><mtext>m</mtext><mo>,</mo><mtext>A</mtext></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mtext>f</mtext><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mtext>A</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mtext>A</mtext><mtext>.</mtext></mrow></math><tex-math>{\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{.}}</tex-math></alternatives></disp-formula>Furthermore, the function F, defined by
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_007.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>A</mtext><mo>,</mo><mtext>B</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtext>n</mtext><mo>∈</mo><msub><mrow><mover><mtext>𝕅</mtext><mo>¯</mo></mover></mrow><mtext>0</mtext></msub><mo>:</mo><mi> </mi><mi> </mi><mtext>A</mtext><mo>⊆</mo><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>,</mo><mtext>B</mtext></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow><mi> </mi><mi> </mi><mtext>for</mtext><mi> </mi><mi> </mi><mtext>A</mtext><mo>,</mo><mtext>B</mtext><mo>⊆</mo><mtext>X</mtext><mo>,</mo></mrow></math><tex-math>{\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,}}</tex-math></alternatives></disp-formula>
satisfies the Sincov problem
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_008.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>A</mtext><mo>,</mo><mtext>B</mtext></mrow><mo>)</mo></mrow><mo>+</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>B</mtext><mo>,</mo><mtext>C</mtext></mrow><mo>)</mo></mrow><mo>⊆</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>A</mtext><mo>,</mo><mtext>C</mtext></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mn>0</mn><mo>∈</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>A</mtext><mo>,</mo><mtext>A</mtext></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>{\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).</tex-math></alternatives></disp-formula>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
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_009.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>+</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>y</mtext><mo>,</mo><mtext>z</mtext></mrow><mo>)</mo></mrow><mo>⊆</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>z</mtext></mrow><mo>)</mo></mrow></mrow></math><tex-math>{\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)</tex-math></alternatives></disp-formula>
for all x, y, z ∈ X. For this, we introduce the convenient notations
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_010.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>R</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>y</mtext><mo>,</mo><mtext>x</mtext></mrow><mo>)</mo></mrow><mi> </mi><mi> </mi><mi> </mi><mtext>and</mtext><mi> </mi><mi> </mi><mtext>S</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>+</mo><mtext>R</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>{\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),</tex-math></alternatives></disp-formula>
and
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0004_eq_011.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>Φ</mi><mrow><mo>(</mo><mtext>x</mtext><mo>)</mo></mrow><mo>=</mo><mtext>F</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>x</mtext></mrow><mo>)</mo></mrow><mi> </mi><mi> </mi><mtext>and</mtext><mi> </mi><mi> </mi><mi>Ψ</mi><mrow><mo>(</mo><mtext>x</mtext><mo>)</mo></mrow><munder><mo>∪</mo><mrow><mtext>y</mtext><mo>∈</mo><mtext>X</mtext></mrow></munder><mrow><mtext>S</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mrow></math><tex-math>\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).}</tex-math></alternatives></disp-formula>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.</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
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◦ϕm⊆ ϕn+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.

Composition iterates, Cauchy, translation, and Sincov inclusions

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"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, Cauchy, translation, and Sincov inclusions

Publié en ligne: 16 juil. 2020

Pages: 54 - 84

Reçu: 17 janv. 2019

DOI: https://doi.org/10.2478/ausm-2020-0004

Mots clés
relations, groupoids, composition iterates, Cauchy, translation and Sincov inclusions

© 2020 W lodzimierz Fechner et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Composition iterates, Cauchy, translation, and Sincov inclusions

Publié en ligne: 16 juil. 2020

Pages: 54 - 84

Reçu: 17 janv. 2019

DOI: https://doi.org/10.2478/ausm-2020-0004

Mots clésrelations, groupoids, composition iterates, Cauchy, translation and Sincov inclusions

© 2020 W lodzimierz Fechner et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
relations, groupoids, composition iterates, Cauchy, translation and Sincov inclusions