Underlying set of the free monoid, does it contain the empty string?
In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element.
Is the empty string an element of the underlying set of the free monoid?
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: $ {ab,aab,a,b...} $ I'm just wondering how exactly is the empty sequence represented in this underlying set?
Take a look at Wikipedia Kleene star:
If V is a set of symbols or characters, then $V*$ is the set of all strings over symbols in $V$, including the empty string $epsilon$.
category-theory monoid forgetful-functors
asked Nov 20 at 10:54
Roland
19511
add a comment |
In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element.
Is the empty string an element of the underlying set of the free monoid?
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: $ {ab,aab,a,b...} $ I'm just wondering how exactly is the empty sequence represented in this underlying set?
Take a look at Wikipedia Kleene star:
If V is a set of symbols or characters, then $V*$ is the set of all strings over symbols in $V$, including the empty string $epsilon$.
category-theory monoid forgetful-functors
asked Nov 20 at 10:54
Roland
19511
add a comment |
In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element.
Is the empty string an element of the underlying set of the free monoid?
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: $ {ab,aab,a,b...} $ I'm just wondering how exactly is the empty sequence represented in this underlying set?
Take a look at Wikipedia Kleene star:
If V is a set of symbols or characters, then $V*$ is the set of all strings over symbols in $V$, including the empty string $epsilon$.
category-theory monoid forgetful-functors
asked Nov 20 at 10:54
Roland
19511
In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element.
Is the empty string an element of the underlying set of the free monoid?
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: $ {ab,aab,a,b...} $ I'm just wondering how exactly is the empty sequence represented in this underlying set?
Take a look at Wikipedia Kleene star:
If V is a set of symbols or characters, then $V*$ is the set of all strings over symbols in $V$, including the empty string $epsilon$.
category-theory monoid forgetful-functors
category-theory monoid forgetful-functors
asked Nov 20 at 10:54
Roland
19511
asked Nov 20 at 10:54
Roland
19511
edited Nov 20 at 14:53
asked Nov 20 at 10:54
Roland
19511
asked Nov 20 at 10:54
Roland
19511
asked Nov 20 at 10:54
Roland
19511
19511
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
A monoid is a tuple $(M,m,e)$ where $M$ is a set, $m:Mtimes M to M$ is an associative law and $ein M$ is a neutral element for $m$.
By definition, the underlying set of $(M,m,e)$ is $M$: therefore $ein M$ means that $e$ is an element of the underlying set.
In the free monoid generated by $A$, the neutral element is the empty string, so the empty string does belong to the underlying set of the free monoid, but there's nothing special about the free monoid here.
answered Nov 20 at 11:03
Max
12.7k11040
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
add a comment |
Yes, it is.
If $M$ is the monoid free over set $S$ then you can identify the elements of $M$ with functions $nto S$ where $n$ is a nonnegative integer with $n:={0,dots,n-1}$.
Then $0$ is the empty set and the empty string is the empty function $0=varnothingto S$.
The empty function is also the empty set, so the empty string corresponds with $varnothing$.
If e.g. we are dealing with function $2to S$ of the form ${(0,a),(1,b)}$ then this function corresponds with string "ab".
answered Nov 20 at 11:19
drhab
97.6k544128
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%2f3006182%2funderlying-set-of-the-free-monoid-does-it-contain-the-empty-string%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
A monoid is a tuple $(M,m,e)$ where $M$ is a set, $m:Mtimes M to M$ is an associative law and $ein M$ is a neutral element for $m$.
By definition, the underlying set of $(M,m,e)$ is $M$: therefore $ein M$ means that $e$ is an element of the underlying set.
In the free monoid generated by $A$, the neutral element is the empty string, so the empty string does belong to the underlying set of the free monoid, but there's nothing special about the free monoid here.
answered Nov 20 at 11:03
Max
12.7k11040
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
add a comment |
A monoid is a tuple $(M,m,e)$ where $M$ is a set, $m:Mtimes M to M$ is an associative law and $ein M$ is a neutral element for $m$.
By definition, the underlying set of $(M,m,e)$ is $M$: therefore $ein M$ means that $e$ is an element of the underlying set.
In the free monoid generated by $A$, the neutral element is the empty string, so the empty string does belong to the underlying set of the free monoid, but there's nothing special about the free monoid here.
answered Nov 20 at 11:03
Max
12.7k11040
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
add a comment |
A monoid is a tuple $(M,m,e)$ where $M$ is a set, $m:Mtimes M to M$ is an associative law and $ein M$ is a neutral element for $m$.
By definition, the underlying set of $(M,m,e)$ is $M$: therefore $ein M$ means that $e$ is an element of the underlying set.
In the free monoid generated by $A$, the neutral element is the empty string, so the empty string does belong to the underlying set of the free monoid, but there's nothing special about the free monoid here.
answered Nov 20 at 11:03
Max
12.7k11040
A monoid is a tuple $(M,m,e)$ where $M$ is a set, $m:Mtimes M to M$ is an associative law and $ein M$ is a neutral element for $m$.
By definition, the underlying set of $(M,m,e)$ is $M$: therefore $ein M$ means that $e$ is an element of the underlying set.
In the free monoid generated by $A$, the neutral element is the empty string, so the empty string does belong to the underlying set of the free monoid, but there's nothing special about the free monoid here.
answered Nov 20 at 11:03
Max
12.7k11040
answered Nov 20 at 11:03
Max
12.7k11040
answered Nov 20 at 11:03
Max
12.7k11040
answered Nov 20 at 11:03
Max
12.7k11040
12.7k11040
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
add a comment |
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: ${ab, aab, a, b}$ I'm just wondering how exactly is the empty sequence represented in this underlying set.
– Roland
Nov 20 at 11:08
2
2
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
@Roland As the empty string. You can name it whatever you like.
– Tobias Kildetoft
Nov 20 at 11:17
3
3
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
@Roland Said another way, "finite" includes length 0!
– Kevin Carlson
Nov 20 at 17:21
add a comment |
Yes, it is.
If $M$ is the monoid free over set $S$ then you can identify the elements of $M$ with functions $nto S$ where $n$ is a nonnegative integer with $n:={0,dots,n-1}$.
Then $0$ is the empty set and the empty string is the empty function $0=varnothingto S$.
The empty function is also the empty set, so the empty string corresponds with $varnothing$.
If e.g. we are dealing with function $2to S$ of the form ${(0,a),(1,b)}$ then this function corresponds with string "ab".
answered Nov 20 at 11:19
drhab
97.6k544128
add a comment |
Yes, it is.
If $M$ is the monoid free over set $S$ then you can identify the elements of $M$ with functions $nto S$ where $n$ is a nonnegative integer with $n:={0,dots,n-1}$.
Then $0$ is the empty set and the empty string is the empty function $0=varnothingto S$.
The empty function is also the empty set, so the empty string corresponds with $varnothing$.
If e.g. we are dealing with function $2to S$ of the form ${(0,a),(1,b)}$ then this function corresponds with string "ab".
answered Nov 20 at 11:19
drhab
97.6k544128
add a comment |
Yes, it is.
If $M$ is the monoid free over set $S$ then you can identify the elements of $M$ with functions $nto S$ where $n$ is a nonnegative integer with $n:={0,dots,n-1}$.
Then $0$ is the empty set and the empty string is the empty function $0=varnothingto S$.
The empty function is also the empty set, so the empty string corresponds with $varnothing$.
If e.g. we are dealing with function $2to S$ of the form ${(0,a),(1,b)}$ then this function corresponds with string "ab".
answered Nov 20 at 11:19
drhab
97.6k544128
Yes, it is.
If $M$ is the monoid free over set $S$ then you can identify the elements of $M$ with functions $nto S$ where $n$ is a nonnegative integer with $n:={0,dots,n-1}$.
Then $0$ is the empty set and the empty string is the empty function $0=varnothingto S$.
The empty function is also the empty set, so the empty string corresponds with $varnothing$.
If e.g. we are dealing with function $2to S$ of the form ${(0,a),(1,b)}$ then this function corresponds with string "ab".
answered Nov 20 at 11:19
drhab
97.6k544128
answered Nov 20 at 11:19
drhab
97.6k544128
answered Nov 20 at 11:19
drhab
97.6k544128
answered Nov 20 at 11:19
drhab
97.6k544128
97.6k544128
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3006182%2funderlying-set-of-the-free-monoid-does-it-contain-the-empty-string%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
