Existence of infimum on the set of equivalence classes
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Let $Q$ be a complete lattice.
Let $sim$ be an equivalence relation on $Q$ conforming to the axiom $$(f_0sim f_1wedge g_0sim g_1wedge f_0leq g_0)Rightarrow f_1leq g_1.$$
Define the order on the set $Q/sim$ of equivalence classes in the obvious way.
Now let $S$ be a (nonempty) set of equivalence classes. Is it true that the infimum $inf S$ necessarily exists and $inf S=[inf_{Xin S} A_X]$ where $A_Xin X$ for every equivalence class $X$?
order-theory equivalence-relations lattice-orders
asked Nov 15 at 18:48
porton
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Let $Q$ be a complete lattice.
Let $sim$ be an equivalence relation on $Q$ conforming to the axiom $$(f_0sim f_1wedge g_0sim g_1wedge f_0leq g_0)Rightarrow f_1leq g_1.$$
Define the order on the set $Q/sim$ of equivalence classes in the obvious way.
Now let $S$ be a (nonempty) set of equivalence classes. Is it true that the infimum $inf S$ necessarily exists and $inf S=[inf_{Xin S} A_X]$ where $A_Xin X$ for every equivalence class $X$?
order-theory equivalence-relations lattice-orders
asked Nov 15 at 18:48
porton
1,86011127
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Q$ be a complete lattice.
Let $sim$ be an equivalence relation on $Q$ conforming to the axiom $$(f_0sim f_1wedge g_0sim g_1wedge f_0leq g_0)Rightarrow f_1leq g_1.$$
Define the order on the set $Q/sim$ of equivalence classes in the obvious way.
Now let $S$ be a (nonempty) set of equivalence classes. Is it true that the infimum $inf S$ necessarily exists and $inf S=[inf_{Xin S} A_X]$ where $A_Xin X$ for every equivalence class $X$?
order-theory equivalence-relations lattice-orders
asked Nov 15 at 18:48
porton
1,86011127
Let $Q$ be a complete lattice.
Let $sim$ be an equivalence relation on $Q$ conforming to the axiom $$(f_0sim f_1wedge g_0sim g_1wedge f_0leq g_0)Rightarrow f_1leq g_1.$$
Define the order on the set $Q/sim$ of equivalence classes in the obvious way.
Now let $S$ be a (nonempty) set of equivalence classes. Is it true that the infimum $inf S$ necessarily exists and $inf S=[inf_{Xin S} A_X]$ where $A_Xin X$ for every equivalence class $X$?
order-theory equivalence-relations lattice-orders
order-theory equivalence-relations lattice-orders
asked Nov 15 at 18:48
porton
1,86011127
asked Nov 15 at 18:48
porton
1,86011127
asked Nov 15 at 18:48
porton
1,86011127
asked Nov 15 at 18:48
porton
1,86011127
asked Nov 15 at 18:48
porton
1,86011127
1,86011127
add a comment |
add a comment |
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$[inf_{Xin S}A_X]leq X$ for every $Xin S$ because $inf_{Xin S}A_Xleq A_X$.
Suppose an equivalence class $Lleq X$ for every $Xin S$.
Take $lin L$. We have $lleq A_X$; $lleqinf_{Xin S}A_X$; $Lleq[inf_{Xin S}A_X]$.
Thus $[inf_{Xin S}A_X]$ is the greatest lower bound of $S$.
answered Nov 15 at 19:17
porton
1,86011127
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
$[inf_{Xin S}A_X]leq X$ for every $Xin S$ because $inf_{Xin S}A_Xleq A_X$.
Suppose an equivalence class $Lleq X$ for every $Xin S$.
Take $lin L$. We have $lleq A_X$; $lleqinf_{Xin S}A_X$; $Lleq[inf_{Xin S}A_X]$.
Thus $[inf_{Xin S}A_X]$ is the greatest lower bound of $S$.
answered Nov 15 at 19:17
porton
1,86011127
add a comment |
up vote
0
down vote
accepted
$[inf_{Xin S}A_X]leq X$ for every $Xin S$ because $inf_{Xin S}A_Xleq A_X$.
Suppose an equivalence class $Lleq X$ for every $Xin S$.
Take $lin L$. We have $lleq A_X$; $lleqinf_{Xin S}A_X$; $Lleq[inf_{Xin S}A_X]$.
Thus $[inf_{Xin S}A_X]$ is the greatest lower bound of $S$.
answered Nov 15 at 19:17
porton
1,86011127
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
$[inf_{Xin S}A_X]leq X$ for every $Xin S$ because $inf_{Xin S}A_Xleq A_X$.
Suppose an equivalence class $Lleq X$ for every $Xin S$.
Take $lin L$. We have $lleq A_X$; $lleqinf_{Xin S}A_X$; $Lleq[inf_{Xin S}A_X]$.
Thus $[inf_{Xin S}A_X]$ is the greatest lower bound of $S$.
answered Nov 15 at 19:17
porton
1,86011127
$[inf_{Xin S}A_X]leq X$ for every $Xin S$ because $inf_{Xin S}A_Xleq A_X$.
Suppose an equivalence class $Lleq X$ for every $Xin S$.
Take $lin L$. We have $lleq A_X$; $lleqinf_{Xin S}A_X$; $Lleq[inf_{Xin S}A_X]$.
Thus $[inf_{Xin S}A_X]$ is the greatest lower bound of $S$.
answered Nov 15 at 19:17
porton
1,86011127
answered Nov 15 at 19:17
porton
1,86011127
answered Nov 15 at 19:17
porton
1,86011127
answered Nov 15 at 19:17
porton
1,86011127
1,86011127
add a comment |
add a comment |
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