From 107d2d0efb0daf0f1ab2eb151fabd5c3fb8c828a Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 5 May 2026 18:26:02 +0100 Subject: [PATCH 1/8] more props added --- draft/draft.tex | 36 +++++++++++++++++++++++++++++------- 1 file changed, 29 insertions(+), 7 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 9124826..1c9decb 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -288,6 +288,7 @@ \newcommand{\gra}{\mathbf{Gra}} \newcommand{\obj}{\mathbf{Obj}} \newcommand{\relar}{\mathbf{R}} +\newcommand{\emre}{\mathbf{L}} \newcommand{\rto}{\mathrel{\tikz{\draw[-{Stealth}] (0,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);}}} \newcommand{\powf}{\mathcal{P}} @@ -2149,16 +2150,37 @@ then at least $(x_1,y_3)$ is not in the behavioural equivalence, while it is in \begin{cor} Assuming that a relator $\relar$ over a functor $F\c\Set\to\Set$ satisfies $\relar(g^\op\comp f)\geq (Fg)^\op\comp Ff$ for every functions $f\c X\to Z$ and $g\c Y\to Z$, then $\hat{\relar}$-bisimilarity from a coalgebra $\alpha\c X\to FX$ to itself is sound and complete, using the axiom of choice. \end{cor} +\subsection{Egli-Milner relator} +\begin{definition} + We call the map $\emre\c\rel\to\rel$ the Egli-Milner $\powf$-relator, whenever for every relation $r\c X\rto Y$ it is defined as follows: + \begin{gather*} + \emre r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\} + \end{gather*} +\end{definition} +Egli-Milner relator is not sound or complete, although its symmetrization is sound and complete. \begin{prop} - Assuming that for a relator $\relar$ over $F\c\Set\to\Set$, $\hat{\relar}$ is difunctionally functorial, then $\relar$ is also difunctionally functorial, and vice-versa. + $\hat{\emre}$-similarity from a coalgebra $(\alpha,X)$ to $(\beta,Y)$ is sound and complete. \end{prop} \begin{proof} - $\hat{\relar}$ being difunctionally functorial means that for every functions $f\c X\to FX$ and $g\c Y\to FY$, we have $\hat{\relar}(g^\op\comp f)=(Fg)^\op\comp Ff$. It is equivalent with the both following conditions being true: - \begin{itemize} - \item ${\relar}(g^\op\comp f)\leq(Fg)^\op\comp Ff$, or $({\relar}(f^\op\comp g))^\op\leq(Fg)^\op\comp Ff$ - \item ${\relar}(g^\op\comp f)\geq(Fg)^\op\comp Ff$ and $({\relar}(f^\op\comp g))^\op\geq(Fg)^\op\comp Ff$ - \end{itemize} - The proof from right to left is obvious. For the other direction we assume that $\hat{\relar}$ is difunctinally functorial. Then we have $\hat{\relar}(g^\op\comp f)\leq(Fg)^\op\comp Ff$ and $\hat{\relar}(g^\op\comp f)\geq(Fg)^\op\comp Ff$. The earlier gives the first item, and the later gives the second item. + \todo{Finish.} +\end{proof} +The symmetrization of the Egli-Milner relator is a Barr-relator. Barr-relators is a generalization of the Egli-Milner relator, where the functor is generalized. + +\begin{prop} + Assuming that $r\c X\rto Y$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then $\appr_{X};\hat{\emre};\appr_{Y}=\appr_{X};\hat{\emre}$ and $\appr_{X};\hat{\emre}=\hat{\emre};\appr_{Y}$. +\end{prop} + +\begin{definition} + A relator over a functor $F$ is a Barr-relator, shown by $\bar{F}$, iff for a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have: + \begin{gather*} + \bar{F}r=F\pi_2\comp(F\pi_1)^\op + \end{gather*} +\end{definition} +\begin{prop} + Assuming that $\relar$ is a relator over $F\c\Set\to\Set$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then the relator that takes $r\c X\rto Y$ to $\appr_{X};\relar r;\appr_{Y}$ is a Barr-relator. +\end{prop} +\begin{proof} + \todo{Finish.} \end{proof} \end{document} From 833d86e0ba2c917646c0764a7ce6892324a82063 Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 5 May 2026 20:40:22 +0100 Subject: [PATCH 2/8] Egli-Milner proof --- draft/draft.tex | 27 +++++++++++++++++++++++++-- 1 file changed, 25 insertions(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 1c9decb..591bd2a 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2162,9 +2162,32 @@ Egli-Milner relator is not sound or complete, although its symmetrization is sou $\hat{\emre}$-similarity from a coalgebra $(\alpha,X)$ to $(\beta,Y)$ is sound and complete. \end{prop} \begin{proof} - \todo{Finish.} + We need to prove that for every functions $f\c X\to Z$ and $g\c Y\to Z$, $\hat{\emre}(g^\op\comp f)=(\powf g)^\op\comp\powf f$. + We have $S\;\hat{\emre}(g^\op\comp f)\;T$ iff $S\;\emre(g^\op\comp f)\;T$ and $T\;\emre(f^\op\comp g)\;S$. Then we have + \begin{align*} + S\;\emre(g^\op\comp f)\;T&\\ + &\iff\forall x\in S,\exists y\in T, x\;g^\op\comp f\; y\\ + &\iff \forall x\in S,\exists y\in T,z\in Z, x\;f\;z\; , \; y\;g\;z, + \end{align*} + and + \begin{align*} + T\;\emre(f^\op\comp g)\;S&\\ + &\iff\forall y\in T,\exists x\in S, y\;f^\op\comp g\; x\\ + &\iff \forall y\in T,\exists x\in S, z\in Z, x\;f\;z\; , \; y\;g\;z. + \end{align*} + It is equivalent with the following: + \begin{gather*} + \forall x\in S,\exists y\in T, f(x)=g(y),\\ + \forall y\in T,\exists x\in S, f(x)=g(y). + \end{gather*} + Equivalently, $Im(f)=Im(g)$, and we call images $U$ that is in $\powf Z$. So, we equivalently have + \begin{align*} + S\;\powf f\;U,\; T\;\powf g\;U&\\ + &\iff S\;\powf f\;U,\; U\;(\powf g)^\op\;T\\ + &\iff S\;(\powf g)^\op\comp\powf f\;T + \end{align*}\qed \end{proof} -The symmetrization of the Egli-Milner relator is a Barr-relator. Barr-relators is a generalization of the Egli-Milner relator, where the functor is generalized. +The symmetrization of the Egli-Milner relator is a Barr-relator. Barr-relator is a generalization of the Egli-Milner relator, where the functor is generalized. \begin{prop} Assuming that $r\c X\rto Y$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then $\appr_{X};\hat{\emre};\appr_{Y}=\appr_{X};\hat{\emre}$ and $\appr_{X};\hat{\emre}=\hat{\emre};\appr_{Y}$. From 03c3abc62437b247810bdff4a100d3695bddc44d Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 5 May 2026 20:50:52 +0100 Subject: [PATCH 3/8] more editting --- draft/draft.tex | 19 ++++++++++++++----- 1 file changed, 14 insertions(+), 5 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 591bd2a..f6e9e5a 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2150,7 +2150,7 @@ then at least $(x_1,y_3)$ is not in the behavioural equivalence, while it is in \begin{cor} Assuming that a relator $\relar$ over a functor $F\c\Set\to\Set$ satisfies $\relar(g^\op\comp f)\geq (Fg)^\op\comp Ff$ for every functions $f\c X\to Z$ and $g\c Y\to Z$, then $\hat{\relar}$-bisimilarity from a coalgebra $\alpha\c X\to FX$ to itself is sound and complete, using the axiom of choice. \end{cor} -\subsection{Egli-Milner relator} +\subsection{Egli-Milner relator and Barr relators} \begin{definition} We call the map $\emre\c\rel\to\rel$ the Egli-Milner $\powf$-relator, whenever for every relation $r\c X\rto Y$ it is defined as follows: \begin{gather*} @@ -2187,20 +2187,29 @@ Egli-Milner relator is not sound or complete, although its symmetrization is sou &\iff S\;(\powf g)^\op\comp\powf f\;T \end{align*}\qed \end{proof} -The symmetrization of the Egli-Milner relator is a Barr-relator. Barr-relator is a generalization of the Egli-Milner relator, where the functor is generalized. \begin{prop} - Assuming that $r\c X\rto Y$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then $\appr_{X};\hat{\emre};\appr_{Y}=\appr_{X};\hat{\emre}$ and $\appr_{X};\hat{\emre}=\hat{\emre};\appr_{Y}$. + Assuming that $r\c X\rto Y$, then $\subseteq;\hat{\emre};\subseteq=\subseteq;\hat{\emre}$ and $\subseteq;\hat{\emre}=\hat{\emre};\subseteq$. \end{prop} +Barr relator is a generalization of the Egli-Milner relator, where the functor is generalized. + \begin{definition} - A relator over a functor $F$ is a Barr-relator, shown by $\bar{F}$, iff for a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have: + A relator over a functor $F$ is a Barr relator, shown by $\bar{F}$, iff for a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have: \begin{gather*} \bar{F}r=F\pi_2\comp(F\pi_1)^\op \end{gather*} \end{definition} + \begin{prop} - Assuming that $\relar$ is a relator over $F\c\Set\to\Set$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then the relator that takes $r\c X\rto Y$ to $\appr_{X};\relar r;\appr_{Y}$ is a Barr-relator. + $\hat{L}$ is a Barr relator. +\end{prop} +\begin{proof} + \todo{Finish.} +\end{proof} + +\begin{prop} + Assuming that $\relar$ is a relator over $F\c\Set\to\Set$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then the relator that takes $r\c X\rto Y$ to $\appr_{X};\relar r;\appr_{Y}$ is a Barr relator. \end{prop} \begin{proof} \todo{Finish.} From 79879548465f7205b52d68bc61473056a277060c Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 5 May 2026 21:16:06 +0100 Subject: [PATCH 4/8] more --- draft/draft.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index f6e9e5a..bc8d70a 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2187,10 +2187,11 @@ Egli-Milner relator is not sound or complete, although its symmetrization is sou &\iff S\;(\powf g)^\op\comp\powf f\;T \end{align*}\qed \end{proof} +For every relation $r\rto X\to Y$ $\emre r=\subseteq\;\emre r=\emre r\;\subseteq=\subseteq;\emre r;\subseteq$. +%\begin{prop} +% Assuming that $r\c X\rto Y$, then $\subseteq;\hat{\emre}r;\subseteq=\subseteq;\hat{\emre}r$ and $\subseteq;\hat{\emre}r=\hat{\emre}r;\subseteq$. +%\end{prop} -\begin{prop} - Assuming that $r\c X\rto Y$, then $\subseteq;\hat{\emre};\subseteq=\subseteq;\hat{\emre}$ and $\subseteq;\hat{\emre}=\hat{\emre};\subseteq$. -\end{prop} Barr relator is a generalization of the Egli-Milner relator, where the functor is generalized. From c2229c444c8bb8b6bd7dafedca0dddb707d4a677 Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 5 May 2026 23:45:00 +0100 Subject: [PATCH 5/8] minor --- draft/draft.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/draft/draft.tex b/draft/draft.tex index bc8d70a..b98329d 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2180,7 +2180,7 @@ Egli-Milner relator is not sound or complete, although its symmetrization is sou \forall x\in S,\exists y\in T, f(x)=g(y),\\ \forall y\in T,\exists x\in S, f(x)=g(y). \end{gather*} - Equivalently, $Im(f)=Im(g)$, and we call images $U$ that is in $\powf Z$. So, we equivalently have + Equivalently, $Im(f\mid_S)=Im(g\mid_T)$, and we call images $U$ that is in $\powf Z$. 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1 deletion(-) diff --git a/draft/.gitignore b/draft/.gitignore index 1147478..7966c40 100644 --- a/draft/.gitignore +++ b/draft/.gitignore @@ -20,7 +20,7 @@ # these rules might exclude image files for figures etc. # *.ps # *.eps -# *.pdf +*.pdf ## Generated if empty string is given at "Please type another file name for output:" .pdf From c62bf13250cb5ffeb2c348742efe8146788db596 Mon Sep 17 00:00:00 2001 From: partowp Date: Wed, 6 May 2026 15:54:12 +0100 Subject: [PATCH 8/8] one-sided barr relator --- draft/draft.tex | 33 +++++++++++++++++++++++++++++++++ 1 file changed, 33 insertions(+) diff --git a/draft/draft.tex b/draft/draft.tex index b98329d..63c887b 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -291,6 +291,7 @@ \newcommand{\emre}{\mathbf{L}} \newcommand{\rto}{\mathrel{\tikz{\draw[-{Stealth}] (0,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);}}} \newcommand{\powf}{\mathcal{P}} +\newcommand{\sappr}{\sqsupseteq} \newcommand{\simeet}{% \mathbin{% @@ -2209,6 +2210,38 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \todo{Finish.} \end{proof} +\begin{definition}[One-sided Barr relator] + A relator over a functor $F$ is a one-sided Barr relator, shown by $\overrightarrow{F}$, iff for a partial order $\appr$ over $F$, a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have: + \begin{gather*} + \overrightarrow{F}r=F\pi_2\comp\sappr\comp(F\pi_1)^\op + \end{gather*} +\end{definition} + +\begin{prop} + For every functor $F\c\Set\to\Set$, the symmetrization of the one-sided Bar relator is equal with the Barr relator. +\end{prop} +\begin{proof} + Where there exist $\pi_1\c A\to X$ and $\pi_2\c A\to Y$ such that $r=\pi_2\comp\pi_1^\op$, we assume that $s \;\hat{\overrightarrow{F}}r\; t$, and we need to show that $s\;F\pi_2\comp(F\pi_1)^\op\;t$. Considering that $r^\op=\pi_1\comp\pi_2^\op$, we have: + \begin{align*} + s \;\hat{\overrightarrow{F}}r\; t&\\ + &\iff s\;\overrightarrow{F}r\;t\qquad\&\qquad s \;(\overrightarrow{F}r^\op)^\op\; t\\ + &\iff s\;F\pi_2\comp\sappr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;(F\pi_1\comp\sappr\comp(F\pi_2)^\op)^\op\;t\\ + &\iff s\;F\pi_2\comp\sappr\comp(F\pi_1)^\op\;t \qquad\&\qquad s\;F\pi_2\comp\appr\comp(F\pi_1)^\op\;t + \end{align*} + Since $F\pi_1$ is a function, then $(F\pi_1)^\op$ is an injective map, so there exist exactly one $w\in FA$ such that $(F\pi_1)^\op(s)=w$, and: + \begin{gather*} + w\;F\pi_2\comp\sappr\; t \qquad\&\qquad w\;F\pi_2\comp\appr\; t + \end{gather*} + And similarly, since $F\pi_2$ is also a function we have exactly one $v\in FA$ such that $(F\pi_2^\op)(t)=v$, and: + \begin{align*} + &w\;\sappr\; v \qquad\&\qquad w\;\appr\; v\\ + \iff&(F\pi_1)^\op(s)\;\sappr\; (F\pi_2^\op)(t) \qquad\&\qquad (F\pi_1)^\op(s)\;\appr\; (F\pi_2^\op)(t)\\ + \iff&(F\pi_1)^\op(s)\;=\; (F\pi_2^\op)(t)\\ + \iff&s\; F\pi_2\comp(F\pi_1)^\op\;t\\ + \iff&s\;\bar{F}r\;t + \end{align*}\qed +\end{proof} + \begin{prop} Assuming that $\relar$ is a relator over $F\c\Set\to\Set$, and $\appr_{X}$ and $\appr_{Y}$ are posets over $FX$ and $FY$ respectively, then the relator that takes $r\c X\rto Y$ to $\appr_{X};\relar r;\appr_{Y}$ is a Barr relator. \end{prop}