Codification of a formal language in set theory.How to define the class of terms of a formal language?Some questions concerning set-theoretic models of first-order theoriesFinding a grammar for a formal languageHow to avoid perceived circularity when defining a formal language?Is it possible to formalize all mathematics in terms of ordinals only?Why is an alphabet a subset of the set of strings that it generates?Formally proving the consistency of a formal theoryFoundations of mathematics , proof theory and analogies with internal mechanism of programming languageA formal language problemAxiomatic formal language theoryConnecting models of Set Theory in Jech's text with Model Theory (pages 161-167)

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Codification of a formal language in set theory.


How to define the class of terms of a formal language?Some questions concerning set-theoretic models of first-order theoriesFinding a grammar for a formal languageHow to avoid perceived circularity when defining a formal language?Is it possible to formalize all mathematics in terms of ordinals only?Why is an alphabet a subset of the set of strings that it generates?Formally proving the consistency of a formal theoryFoundations of mathematics , proof theory and analogies with internal mechanism of programming languageA formal language problemAxiomatic formal language theoryConnecting models of Set Theory in Jech's text with Model Theory (pages 161-167)













3












$begingroup$


Starting with an arbitrary class of sets $Gamma$, can you generate a free semigroup $Gamma^*$ over $Gamma$ with the group operation of concatenation ($frown$)?



The goal here is to codify a formal language in terms of set theory.




The difficulty is in coming up with a set-theoretic operation that corresponds to concatenation such that it makes every new element resulting from concatenation unique, and is associative.



Given $a,b in Gamma$, the first thought would be to represent $a frown bfrown c$ as a 3-tuple $<a,b,c>$. I know I can define tuples set-theoretically via $<a,b>:=a,a,b$ but this will violate associativity in concatenation:



$$a frown(b frown c)=<a,<b,c>> ne <<a,b>,c>=(a frown b)frown c$$




I have tried other variants but I haven't been able to come up with a set-theoretic description of concatenation that respects associativity, any ideas?



EDIT: This is a related question: https://mathoverflow.net/questions/12190/set-theoretic-foundations-for-formal-language-theory



unfortunately none of the answers provide an explicit definition of concatenation in set-theoretic terms.










share|cite|improve this question











$endgroup$











  • $begingroup$
    I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
    $endgroup$
    – Derek Elkins
    Mar 10 at 21:16










  • $begingroup$
    @DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
    $endgroup$
    – Mike
    Mar 10 at 21:19






  • 1




    $begingroup$
    Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
    $endgroup$
    – Asaf Karagila
    Mar 10 at 21:19










  • $begingroup$
    @AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
    $endgroup$
    – Mike
    Mar 10 at 21:22






  • 1




    $begingroup$
    Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
    $endgroup$
    – Asaf Karagila
    Mar 11 at 0:01















3












$begingroup$


Starting with an arbitrary class of sets $Gamma$, can you generate a free semigroup $Gamma^*$ over $Gamma$ with the group operation of concatenation ($frown$)?



The goal here is to codify a formal language in terms of set theory.




The difficulty is in coming up with a set-theoretic operation that corresponds to concatenation such that it makes every new element resulting from concatenation unique, and is associative.



Given $a,b in Gamma$, the first thought would be to represent $a frown bfrown c$ as a 3-tuple $<a,b,c>$. I know I can define tuples set-theoretically via $<a,b>:=a,a,b$ but this will violate associativity in concatenation:



$$a frown(b frown c)=<a,<b,c>> ne <<a,b>,c>=(a frown b)frown c$$




I have tried other variants but I haven't been able to come up with a set-theoretic description of concatenation that respects associativity, any ideas?



EDIT: This is a related question: https://mathoverflow.net/questions/12190/set-theoretic-foundations-for-formal-language-theory



unfortunately none of the answers provide an explicit definition of concatenation in set-theoretic terms.










share|cite|improve this question











$endgroup$











  • $begingroup$
    I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
    $endgroup$
    – Derek Elkins
    Mar 10 at 21:16










  • $begingroup$
    @DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
    $endgroup$
    – Mike
    Mar 10 at 21:19






  • 1




    $begingroup$
    Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
    $endgroup$
    – Asaf Karagila
    Mar 10 at 21:19










  • $begingroup$
    @AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
    $endgroup$
    – Mike
    Mar 10 at 21:22






  • 1




    $begingroup$
    Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
    $endgroup$
    – Asaf Karagila
    Mar 11 at 0:01













3












3








3


1



$begingroup$


Starting with an arbitrary class of sets $Gamma$, can you generate a free semigroup $Gamma^*$ over $Gamma$ with the group operation of concatenation ($frown$)?



The goal here is to codify a formal language in terms of set theory.




The difficulty is in coming up with a set-theoretic operation that corresponds to concatenation such that it makes every new element resulting from concatenation unique, and is associative.



Given $a,b in Gamma$, the first thought would be to represent $a frown bfrown c$ as a 3-tuple $<a,b,c>$. I know I can define tuples set-theoretically via $<a,b>:=a,a,b$ but this will violate associativity in concatenation:



$$a frown(b frown c)=<a,<b,c>> ne <<a,b>,c>=(a frown b)frown c$$




I have tried other variants but I haven't been able to come up with a set-theoretic description of concatenation that respects associativity, any ideas?



EDIT: This is a related question: https://mathoverflow.net/questions/12190/set-theoretic-foundations-for-formal-language-theory



unfortunately none of the answers provide an explicit definition of concatenation in set-theoretic terms.










share|cite|improve this question











$endgroup$




Starting with an arbitrary class of sets $Gamma$, can you generate a free semigroup $Gamma^*$ over $Gamma$ with the group operation of concatenation ($frown$)?



The goal here is to codify a formal language in terms of set theory.




The difficulty is in coming up with a set-theoretic operation that corresponds to concatenation such that it makes every new element resulting from concatenation unique, and is associative.



Given $a,b in Gamma$, the first thought would be to represent $a frown bfrown c$ as a 3-tuple $<a,b,c>$. I know I can define tuples set-theoretically via $<a,b>:=a,a,b$ but this will violate associativity in concatenation:



$$a frown(b frown c)=<a,<b,c>> ne <<a,b>,c>=(a frown b)frown c$$




I have tried other variants but I haven't been able to come up with a set-theoretic description of concatenation that respects associativity, any ideas?



EDIT: This is a related question: https://mathoverflow.net/questions/12190/set-theoretic-foundations-for-formal-language-theory



unfortunately none of the answers provide an explicit definition of concatenation in set-theoretic terms.







logic set-theory formal-languages formal-systems formal-grammar






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 10 at 21:19







Mike

















asked Mar 10 at 20:28









MikeMike

759415




759415











  • $begingroup$
    I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
    $endgroup$
    – Derek Elkins
    Mar 10 at 21:16










  • $begingroup$
    @DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
    $endgroup$
    – Mike
    Mar 10 at 21:19






  • 1




    $begingroup$
    Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
    $endgroup$
    – Asaf Karagila
    Mar 10 at 21:19










  • $begingroup$
    @AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
    $endgroup$
    – Mike
    Mar 10 at 21:22






  • 1




    $begingroup$
    Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
    $endgroup$
    – Asaf Karagila
    Mar 11 at 0:01
















  • $begingroup$
    I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
    $endgroup$
    – Derek Elkins
    Mar 10 at 21:16










  • $begingroup$
    @DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
    $endgroup$
    – Mike
    Mar 10 at 21:19






  • 1




    $begingroup$
    Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
    $endgroup$
    – Asaf Karagila
    Mar 10 at 21:19










  • $begingroup$
    @AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
    $endgroup$
    – Mike
    Mar 10 at 21:22






  • 1




    $begingroup$
    Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
    $endgroup$
    – Asaf Karagila
    Mar 11 at 0:01















$begingroup$
I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
$endgroup$
– Derek Elkins
Mar 10 at 21:16




$begingroup$
I'm pretty sure you're going to have issues avoiding both the Axiom of Infinity and the Axiom of Power Sets. Many concepts in formal language theory are inductively defined. There are two approaches to characterizing inductively defined sets: we can say that they are the smallest sets satisfying some condition, or we can that they are the union of an (countably) infinite number of "stages". The former is impredicative and thus requires something like powersets, while the latter relies the existence of naturals.
$endgroup$
– Derek Elkins
Mar 10 at 21:16












$begingroup$
@DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
$endgroup$
– Mike
Mar 10 at 21:19




$begingroup$
@DerekElkins Just to make the task easier, let me remove that restriction from the question. If someone can provide insight about the definition of concatenation in terms of set theory at this point I'll be happy no matter what axioms of set theory they assume.
$endgroup$
– Mike
Mar 10 at 21:19




1




1




$begingroup$
Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
$endgroup$
– Asaf Karagila
Mar 10 at 21:19




$begingroup$
Foundation has nothing to do with that. Neither does infinity nor power set. You just need enough to prove there are infinitely many ordinals, then you can isolate the finite ones (no need for any of the aforementioned axioms). Then just define the sequences in the obvious way.
$endgroup$
– Asaf Karagila
Mar 10 at 21:19












$begingroup$
@AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
$endgroup$
– Mike
Mar 10 at 21:22




$begingroup$
@AsafKaragila You mean define $a frown b frown c$ as the sequence $a,b,c$? Wouldn't that cause the same problem with associativity that the tuples cause in my question?
$endgroup$
– Mike
Mar 10 at 21:22




1




1




$begingroup$
Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
$endgroup$
– Asaf Karagila
Mar 11 at 0:01




$begingroup$
Mike, I've posted an answer there by the time I saw this comment. Do note that it might be very helpful to understand how PA encodes first-order logic, since this is almost the same thing.
$endgroup$
– Asaf Karagila
Mar 11 at 0:01










1 Answer
1






active

oldest

votes


















3












$begingroup$

There are a couple ways to address this. Equivalence classes give the most algebraically natural treatment:



  • Taking the naive definition of concatenation-as-ordered-pair, we get the free magma $hatGamma$ on $Gamma$.



  • Now the free semigroup will just be the free magma modulo the "associativity relation" - basically, we just need to whip up the binary relation $sim$ describing when two elements of $hatGamma$ "should be" equal. It's a bit messy to describe $sim$ "explicitly" - this winds up being an inductive construction - but we can also define it as the smallest equivalence relation on $hatGamma$ such that (or, the intersection of all equivalence relations on $hatGamma$ such that) for all $a,b,c,dinhatGamma$ we have:



    • $asim b$ and $csim d$ implies $langle a,cranglesimlangle b,drangle$, and


    • $langle a,langle b,crangleranglesim langlelangle a,brangle, crangle$.



It's now easy to define the semigroup operation $cdot$ as $$[a]_simcdot[b]_sim=[langle a,brangle_sim$$ (after, of course, checking that this is in fact well-defined). Or if you want to be really pedantic about it, given $sim$-classes $E,F$, their product $Ecdot F$ is the unique $sim$-class $G$ such that there are elements $ain E$ and $bin F$ such that $langle a,branglein G$.




Another approach, less algebraically natural but perhaps more concrete, is via tuples as functions.



Specifically:



  • An element of $Gamma^*$ will be a function $f$ such that $(1)$ the domain of $f$ is some natural number $n$, and $(2)$ the range of $f$ is $subseteqGamma$. A function with domain $n$ is "morally" an $n$-tuple.



  • We can now define a fully associative version of concatenation using arithmetic. Specifically, suppose $f,gin Gamma^*$ with domains $m,n$ respectively. We let $f^smallfrown g$ be the function $h$ given by: $dom(h)=m+n$, for $k<m$ we have $h(k)=f(k)$, and for $mle k<n$ we have $h(k)=g(k-m)$.



    • So for example if $dom(f)=2, dom(g)=1$, $f$ sends $0$ to $0$ and $1$ to $1$, and $g$ sends $0$ to $3$, then $f^smallfrown g$ has domain $3$, sends $0$ to $0$, sends $1$ to $1$, and sends $2$ to $3$.


(Remember that in set theory a natural number is just a finite ordinal, and in particular is just the set of smaller natural number; so e.g. "$dom(f)=5$" makes perfect sense.)



The only thing this relies on is arithmetic of finite ordinals, which is straightforward to develop.






share|cite|improve this answer











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    3












    $begingroup$

    There are a couple ways to address this. Equivalence classes give the most algebraically natural treatment:



    • Taking the naive definition of concatenation-as-ordered-pair, we get the free magma $hatGamma$ on $Gamma$.



    • Now the free semigroup will just be the free magma modulo the "associativity relation" - basically, we just need to whip up the binary relation $sim$ describing when two elements of $hatGamma$ "should be" equal. It's a bit messy to describe $sim$ "explicitly" - this winds up being an inductive construction - but we can also define it as the smallest equivalence relation on $hatGamma$ such that (or, the intersection of all equivalence relations on $hatGamma$ such that) for all $a,b,c,dinhatGamma$ we have:



      • $asim b$ and $csim d$ implies $langle a,cranglesimlangle b,drangle$, and


      • $langle a,langle b,crangleranglesim langlelangle a,brangle, crangle$.



    It's now easy to define the semigroup operation $cdot$ as $$[a]_simcdot[b]_sim=[langle a,brangle_sim$$ (after, of course, checking that this is in fact well-defined). Or if you want to be really pedantic about it, given $sim$-classes $E,F$, their product $Ecdot F$ is the unique $sim$-class $G$ such that there are elements $ain E$ and $bin F$ such that $langle a,branglein G$.




    Another approach, less algebraically natural but perhaps more concrete, is via tuples as functions.



    Specifically:



    • An element of $Gamma^*$ will be a function $f$ such that $(1)$ the domain of $f$ is some natural number $n$, and $(2)$ the range of $f$ is $subseteqGamma$. A function with domain $n$ is "morally" an $n$-tuple.



    • We can now define a fully associative version of concatenation using arithmetic. Specifically, suppose $f,gin Gamma^*$ with domains $m,n$ respectively. We let $f^smallfrown g$ be the function $h$ given by: $dom(h)=m+n$, for $k<m$ we have $h(k)=f(k)$, and for $mle k<n$ we have $h(k)=g(k-m)$.



      • So for example if $dom(f)=2, dom(g)=1$, $f$ sends $0$ to $0$ and $1$ to $1$, and $g$ sends $0$ to $3$, then $f^smallfrown g$ has domain $3$, sends $0$ to $0$, sends $1$ to $1$, and sends $2$ to $3$.


    (Remember that in set theory a natural number is just a finite ordinal, and in particular is just the set of smaller natural number; so e.g. "$dom(f)=5$" makes perfect sense.)



    The only thing this relies on is arithmetic of finite ordinals, which is straightforward to develop.






    share|cite|improve this answer











    $endgroup$

















      3












      $begingroup$

      There are a couple ways to address this. Equivalence classes give the most algebraically natural treatment:



      • Taking the naive definition of concatenation-as-ordered-pair, we get the free magma $hatGamma$ on $Gamma$.



      • Now the free semigroup will just be the free magma modulo the "associativity relation" - basically, we just need to whip up the binary relation $sim$ describing when two elements of $hatGamma$ "should be" equal. It's a bit messy to describe $sim$ "explicitly" - this winds up being an inductive construction - but we can also define it as the smallest equivalence relation on $hatGamma$ such that (or, the intersection of all equivalence relations on $hatGamma$ such that) for all $a,b,c,dinhatGamma$ we have:



        • $asim b$ and $csim d$ implies $langle a,cranglesimlangle b,drangle$, and


        • $langle a,langle b,crangleranglesim langlelangle a,brangle, crangle$.



      It's now easy to define the semigroup operation $cdot$ as $$[a]_simcdot[b]_sim=[langle a,brangle_sim$$ (after, of course, checking that this is in fact well-defined). Or if you want to be really pedantic about it, given $sim$-classes $E,F$, their product $Ecdot F$ is the unique $sim$-class $G$ such that there are elements $ain E$ and $bin F$ such that $langle a,branglein G$.




      Another approach, less algebraically natural but perhaps more concrete, is via tuples as functions.



      Specifically:



      • An element of $Gamma^*$ will be a function $f$ such that $(1)$ the domain of $f$ is some natural number $n$, and $(2)$ the range of $f$ is $subseteqGamma$. A function with domain $n$ is "morally" an $n$-tuple.



      • We can now define a fully associative version of concatenation using arithmetic. Specifically, suppose $f,gin Gamma^*$ with domains $m,n$ respectively. We let $f^smallfrown g$ be the function $h$ given by: $dom(h)=m+n$, for $k<m$ we have $h(k)=f(k)$, and for $mle k<n$ we have $h(k)=g(k-m)$.



        • So for example if $dom(f)=2, dom(g)=1$, $f$ sends $0$ to $0$ and $1$ to $1$, and $g$ sends $0$ to $3$, then $f^smallfrown g$ has domain $3$, sends $0$ to $0$, sends $1$ to $1$, and sends $2$ to $3$.


      (Remember that in set theory a natural number is just a finite ordinal, and in particular is just the set of smaller natural number; so e.g. "$dom(f)=5$" makes perfect sense.)



      The only thing this relies on is arithmetic of finite ordinals, which is straightforward to develop.






      share|cite|improve this answer











      $endgroup$















        3












        3








        3





        $begingroup$

        There are a couple ways to address this. Equivalence classes give the most algebraically natural treatment:



        • Taking the naive definition of concatenation-as-ordered-pair, we get the free magma $hatGamma$ on $Gamma$.



        • Now the free semigroup will just be the free magma modulo the "associativity relation" - basically, we just need to whip up the binary relation $sim$ describing when two elements of $hatGamma$ "should be" equal. It's a bit messy to describe $sim$ "explicitly" - this winds up being an inductive construction - but we can also define it as the smallest equivalence relation on $hatGamma$ such that (or, the intersection of all equivalence relations on $hatGamma$ such that) for all $a,b,c,dinhatGamma$ we have:



          • $asim b$ and $csim d$ implies $langle a,cranglesimlangle b,drangle$, and


          • $langle a,langle b,crangleranglesim langlelangle a,brangle, crangle$.



        It's now easy to define the semigroup operation $cdot$ as $$[a]_simcdot[b]_sim=[langle a,brangle_sim$$ (after, of course, checking that this is in fact well-defined). Or if you want to be really pedantic about it, given $sim$-classes $E,F$, their product $Ecdot F$ is the unique $sim$-class $G$ such that there are elements $ain E$ and $bin F$ such that $langle a,branglein G$.




        Another approach, less algebraically natural but perhaps more concrete, is via tuples as functions.



        Specifically:



        • An element of $Gamma^*$ will be a function $f$ such that $(1)$ the domain of $f$ is some natural number $n$, and $(2)$ the range of $f$ is $subseteqGamma$. A function with domain $n$ is "morally" an $n$-tuple.



        • We can now define a fully associative version of concatenation using arithmetic. Specifically, suppose $f,gin Gamma^*$ with domains $m,n$ respectively. We let $f^smallfrown g$ be the function $h$ given by: $dom(h)=m+n$, for $k<m$ we have $h(k)=f(k)$, and for $mle k<n$ we have $h(k)=g(k-m)$.



          • So for example if $dom(f)=2, dom(g)=1$, $f$ sends $0$ to $0$ and $1$ to $1$, and $g$ sends $0$ to $3$, then $f^smallfrown g$ has domain $3$, sends $0$ to $0$, sends $1$ to $1$, and sends $2$ to $3$.


        (Remember that in set theory a natural number is just a finite ordinal, and in particular is just the set of smaller natural number; so e.g. "$dom(f)=5$" makes perfect sense.)



        The only thing this relies on is arithmetic of finite ordinals, which is straightforward to develop.






        share|cite|improve this answer











        $endgroup$



        There are a couple ways to address this. Equivalence classes give the most algebraically natural treatment:



        • Taking the naive definition of concatenation-as-ordered-pair, we get the free magma $hatGamma$ on $Gamma$.



        • Now the free semigroup will just be the free magma modulo the "associativity relation" - basically, we just need to whip up the binary relation $sim$ describing when two elements of $hatGamma$ "should be" equal. It's a bit messy to describe $sim$ "explicitly" - this winds up being an inductive construction - but we can also define it as the smallest equivalence relation on $hatGamma$ such that (or, the intersection of all equivalence relations on $hatGamma$ such that) for all $a,b,c,dinhatGamma$ we have:



          • $asim b$ and $csim d$ implies $langle a,cranglesimlangle b,drangle$, and


          • $langle a,langle b,crangleranglesim langlelangle a,brangle, crangle$.



        It's now easy to define the semigroup operation $cdot$ as $$[a]_simcdot[b]_sim=[langle a,brangle_sim$$ (after, of course, checking that this is in fact well-defined). Or if you want to be really pedantic about it, given $sim$-classes $E,F$, their product $Ecdot F$ is the unique $sim$-class $G$ such that there are elements $ain E$ and $bin F$ such that $langle a,branglein G$.




        Another approach, less algebraically natural but perhaps more concrete, is via tuples as functions.



        Specifically:



        • An element of $Gamma^*$ will be a function $f$ such that $(1)$ the domain of $f$ is some natural number $n$, and $(2)$ the range of $f$ is $subseteqGamma$. A function with domain $n$ is "morally" an $n$-tuple.



        • We can now define a fully associative version of concatenation using arithmetic. Specifically, suppose $f,gin Gamma^*$ with domains $m,n$ respectively. We let $f^smallfrown g$ be the function $h$ given by: $dom(h)=m+n$, for $k<m$ we have $h(k)=f(k)$, and for $mle k<n$ we have $h(k)=g(k-m)$.



          • So for example if $dom(f)=2, dom(g)=1$, $f$ sends $0$ to $0$ and $1$ to $1$, and $g$ sends $0$ to $3$, then $f^smallfrown g$ has domain $3$, sends $0$ to $0$, sends $1$ to $1$, and sends $2$ to $3$.


        (Remember that in set theory a natural number is just a finite ordinal, and in particular is just the set of smaller natural number; so e.g. "$dom(f)=5$" makes perfect sense.)



        The only thing this relies on is arithmetic of finite ordinals, which is straightforward to develop.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 10 at 21:56

























        answered Mar 10 at 21:46









        Noah SchweberNoah Schweber

        127k10151290




        127k10151290



























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