Partial ordering on a space of matrices
$begingroup$
I am wondering as how to define partial ordering on a space of matrices! The following is what i intuitively constructed:
If $M$ is a set all $n×n$ matrices with entries of each matrix are from an idempotent semiring then $M$ is also an idempptent semiring under the matrix addition and multiplication. Now, we can define a natural order on M as $Aleq B$ when $A+B=B$ for all $A, B$ in $M$, which is a partial order relation on $M$. Is this intuition correct?
matrices order-theory semiring
asked Dec 16 '18 at 17:04
getegete
847
$endgroup$
add a comment |
$begingroup$
I am wondering as how to define partial ordering on a space of matrices! The following is what i intuitively constructed:
If $M$ is a set all $n×n$ matrices with entries of each matrix are from an idempotent semiring then $M$ is also an idempptent semiring under the matrix addition and multiplication. Now, we can define a natural order on M as $Aleq B$ when $A+B=B$ for all $A, B$ in $M$, which is a partial order relation on $M$. Is this intuition correct?
matrices order-theory semiring
asked Dec 16 '18 at 17:04
getegete
847
$endgroup$
add a comment |
$begingroup$
I am wondering as how to define partial ordering on a space of matrices! The following is what i intuitively constructed:
If $M$ is a set all $n×n$ matrices with entries of each matrix are from an idempotent semiring then $M$ is also an idempptent semiring under the matrix addition and multiplication. Now, we can define a natural order on M as $Aleq B$ when $A+B=B$ for all $A, B$ in $M$, which is a partial order relation on $M$. Is this intuition correct?
matrices order-theory semiring
asked Dec 16 '18 at 17:04
getegete
847
$endgroup$
I am wondering as how to define partial ordering on a space of matrices! The following is what i intuitively constructed:
If $M$ is a set all $n×n$ matrices with entries of each matrix are from an idempotent semiring then $M$ is also an idempptent semiring under the matrix addition and multiplication. Now, we can define a natural order on M as $Aleq B$ when $A+B=B$ for all $A, B$ in $M$, which is a partial order relation on $M$. Is this intuition correct?
matrices order-theory semiring
matrices order-theory semiring
asked Dec 16 '18 at 17:04
getegete
847
asked Dec 16 '18 at 17:04
getegete
847
edited Dec 17 '18 at 5:53
gete
asked Dec 16 '18 at 17:04
getegete
847
asked Dec 16 '18 at 17:04
getegete
847
asked Dec 16 '18 at 17:04
getegete
847
847
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A semiring $R$ for all $ain R$, $a + a = a$ can be ordered by $a leq b$ when $a + b = b$.
In addition $leq$ produces a ring order. $a leq b$ implies $a + c leq a + c$, $ac leq bc$.
Have you worked out the details?
The same order can be imposed upon M except possibly for $A leq B$ implies $AC leq BC$, which I haven't looked into.
edited Dec 17 '18 at 5:13
user10354138
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
$endgroup$
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%2f3042845%2fpartial-ordering-on-a-space-of-matrices%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
$begingroup$
A semiring $R$ for all $ain R$, $a + a = a$ can be ordered by $a leq b$ when $a + b = b$.
In addition $leq$ produces a ring order. $a leq b$ implies $a + c leq a + c$, $ac leq bc$.
Have you worked out the details?
The same order can be imposed upon M except possibly for $A leq B$ implies $AC leq BC$, which I haven't looked into.
edited Dec 17 '18 at 5:13
user10354138
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
$endgroup$
add a comment |
$begingroup$
A semiring $R$ for all $ain R$, $a + a = a$ can be ordered by $a leq b$ when $a + b = b$.
In addition $leq$ produces a ring order. $a leq b$ implies $a + c leq a + c$, $ac leq bc$.
Have you worked out the details?
The same order can be imposed upon M except possibly for $A leq B$ implies $AC leq BC$, which I haven't looked into.
edited Dec 17 '18 at 5:13
user10354138
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
$endgroup$
add a comment |
$begingroup$
A semiring $R$ for all $ain R$, $a + a = a$ can be ordered by $a leq b$ when $a + b = b$.
In addition $leq$ produces a ring order. $a leq b$ implies $a + c leq a + c$, $ac leq bc$.
Have you worked out the details?
The same order can be imposed upon M except possibly for $A leq B$ implies $AC leq BC$, which I haven't looked into.
edited Dec 17 '18 at 5:13
user10354138
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
$endgroup$
A semiring $R$ for all $ain R$, $a + a = a$ can be ordered by $a leq b$ when $a + b = b$.
In addition $leq$ produces a ring order. $a leq b$ implies $a + c leq a + c$, $ac leq bc$.
Have you worked out the details?
The same order can be imposed upon M except possibly for $A leq B$ implies $AC leq BC$, which I haven't looked into.
edited Dec 17 '18 at 5:13
user10354138
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
edited Dec 17 '18 at 5:13
user10354138
7,4322925
edited Dec 17 '18 at 5:13
user10354138
7,4322925
edited Dec 17 '18 at 5:13
user10354138
7,4322925
7,4322925
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
answered Dec 17 '18 at 1:25
William ElliotWilliam Elliot
8,6672720
8,6672720
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%2f3042845%2fpartial-ordering-on-a-space-of-matrices%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
