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LSpice
LSpice
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Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{a\in A\mid (x,a)\in S\}\triangle\{a\in A\mid (y,a)\in S\}$$\{a\in A\mid (x,a)\in S\}\mathbin\triangle\{a\in A\mid (y,a)\in S\}$ and $\{a\in A\mid (a,x)\in S\}\triangle\{a\in A\mid (a,y)\in S\}$$\{a\in A\mid (a,x)\in S\}\mathbin\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{a\in A\mid (x,a)\in S\}$ and $\{a\in A\mid (a,y)\in S\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{a\in A\mid (x,a)\in S\}\triangle\{a\in A\mid (y,a)\in S\}$ and $\{a\in A\mid (a,x)\in S\}\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{a\in A\mid (x,a)\in S\}$ and $\{a\in A\mid (a,y)\in S\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{a\in A\mid (x,a)\in S\}\mathbin\triangle\{a\in A\mid (y,a)\in S\}$ and $\{a\in A\mid (a,x)\in S\}\mathbin\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{a\in A\mid (x,a)\in S\}$ and $\{a\in A\mid (a,y)\in S\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

added 18 characters in body
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Markiian Khylynskyi
Markiian Khylynskyi
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Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{(a,b)\in S\mid a=x\}\triangle\{(a,b)\in S\mid a=y\}$$\{a\in A\mid (x,a)\in S\}\triangle\{a\in A\mid (y,a)\in S\}$ and $\{(a,b)\in S\mid b=x\}\triangle\{(a,b)\in S\mid b=y\}$$\{a\in A\mid (a,x)\in S\}\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{(a,b)\in S\mid a=x\}$$\{a\in A\mid (x,a)\in S\}$ and $\{(a,b)\in S\mid b=y\}$$\{a\in A\mid (a,y)\in S\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{(a,b)\in S\mid a=x\}\triangle\{(a,b)\in S\mid a=y\}$ and $\{(a,b)\in S\mid b=x\}\triangle\{(a,b)\in S\mid b=y\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{(a,b)\in S\mid a=x\}$ and $\{(a,b)\in S\mid b=y\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{a\in A\mid (x,a)\in S\}\triangle\{a\in A\mid (y,a)\in S\}$ and $\{a\in A\mid (a,x)\in S\}\triangle\{a\in A\mid (a,y)\in S\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{a\in A\mid (x,a)\in S\}$ and $\{a\in A\mid (a,y)\in S\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

deleted 12 characters in body
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Markiian Khylynskyi
Markiian Khylynskyi
  • 134
  • 7

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{(a,c)\mid (a,c)\in S\}\triangle\{(b,c)\mid (b,c)\in S\}$$\{(a,b)\in S\mid a=x\}\triangle\{(a,b)\in S\mid a=y\}$ and $\{(c,a)\mid (c,a)\in S\}\triangle\{(c,b)\mid (c,b)\in S\}$$\{(a,b)\in S\mid b=x\}\triangle\{(a,b)\in S\mid b=y\}$ are finite for any $a,b\in A$$x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{(a,c)\mid (a,c)\in S\}$$\{(a,b)\in S\mid a=x\}$ and $\{(c,b)\mid (c,b)\in S\}$$\{(a,b)\in S\mid b=y\}$ are finite (cofinite) for any $a,b\in A$$x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{(a,c)\mid (a,c)\in S\}\triangle\{(b,c)\mid (b,c)\in S\}$ and $\{(c,a)\mid (c,a)\in S\}\triangle\{(c,b)\mid (c,b)\in S\}$ are finite for any $a,b\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{(a,c)\mid (a,c)\in S\}$ and $\{(c,b)\mid (c,b)\in S\}$ are finite (cofinite) for any $a,b\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

Let $A$ be any set and $S$ be a subset of $A^2$ such that sets $\{(a,b)\in S\mid a=x\}\triangle\{(a,b)\in S\mid a=y\}$ and $\{(a,b)\in S\mid b=x\}\triangle\{(a,b)\in S\mid b=y\}$ are finite for any $x,y\in A$. What can be the set $S$?

There are two simple examples of the set $S$, when sets $\{(a,b)\in S\mid a=x\}$ and $\{(a,b)\in S\mid b=y\}$ are finite (cofinite) for any $x,y\in A$. If $A=\mathbb{Z}$, then $S$ can be $\{(a,b)\in\mathbb{Z}^2\mid a < b\}$. Are there any other examples?

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Markiian Khylynskyi
Markiian Khylynskyi
  • 134
  • 7