Mixing
How to prove that for any nonempty set $A$, there exists a maximal partial order $lambda$ with the axiom of...
$begingroup$
The ``maximal partial order'' means for any partial order $alphainmathscr{B}(A)$, $lambdasubseteqalpha$ implies $lambda=alpha$.
It is obvious if we apply the well-ordering theorem or Zorn's lemma. Since the equivalence among well-ordering theorem, Zorn's lemma and the axiom of choice, we can give a proof directly from the axiom of choice, I mean, not a pretended proof( to prove Zorn's lemma first and then use the lemma).
Could any one help me?
order-theory
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
$endgroup$
add a comment |
$begingroup$
The ``maximal partial order'' means for any partial order $alphainmathscr{B}(A)$, $lambdasubseteqalpha$ implies $lambda=alpha$.
It is obvious if we apply the well-ordering theorem or Zorn's lemma. Since the equivalence among well-ordering theorem, Zorn's lemma and the axiom of choice, we can give a proof directly from the axiom of choice, I mean, not a pretended proof( to prove Zorn's lemma first and then use the lemma).
Could any one help me?
order-theory
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
$endgroup$
add a comment |
$begingroup$
The ``maximal partial order'' means for any partial order $alphainmathscr{B}(A)$, $lambdasubseteqalpha$ implies $lambda=alpha$.
It is obvious if we apply the well-ordering theorem or Zorn's lemma. Since the equivalence among well-ordering theorem, Zorn's lemma and the axiom of choice, we can give a proof directly from the axiom of choice, I mean, not a pretended proof( to prove Zorn's lemma first and then use the lemma).
Could any one help me?
order-theory
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
$endgroup$
The ``maximal partial order'' means for any partial order $alphainmathscr{B}(A)$, $lambdasubseteqalpha$ implies $lambda=alpha$.
It is obvious if we apply the well-ordering theorem or Zorn's lemma. Since the equivalence among well-ordering theorem, Zorn's lemma and the axiom of choice, we can give a proof directly from the axiom of choice, I mean, not a pretended proof( to prove Zorn's lemma first and then use the lemma).
Could any one help me?
order-theory
order-theory
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
asked Jan 21 at 6:36
闫嘉琦闫嘉琦
648112
648112
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Apply Zorn's lemma to the collection $mathcal{C}$ of partial orders in $A$. Here $mathcal{C}$ is ordered by $subseteq$.
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3081568%2fhow-to-prove-that-for-any-nonempty-set-a-there-exists-a-maximal-partial-order%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Apply Zorn's lemma to the collection $mathcal{C}$ of partial orders in $A$. Here $mathcal{C}$ is ordered by $subseteq$.
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
$endgroup$
add a comment |
$begingroup$
Apply Zorn's lemma to the collection $mathcal{C}$ of partial orders in $A$. Here $mathcal{C}$ is ordered by $subseteq$.
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
$endgroup$
add a comment |
$begingroup$
Apply Zorn's lemma to the collection $mathcal{C}$ of partial orders in $A$. Here $mathcal{C}$ is ordered by $subseteq$.
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
$endgroup$
Apply Zorn's lemma to the collection $mathcal{C}$ of partial orders in $A$. Here $mathcal{C}$ is ordered by $subseteq$.
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
answered Jan 30 at 19:52
Alberto TakaseAlberto Takase
2,310719
2,310719
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3081568%2fhow-to-prove-that-for-any-nonempty-set-a-there-exists-a-maximal-partial-order%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
