Changes

^{\backprime\backprime}\operatorname{blank}^{\prime\prime}

^{\backprime\backprime}\operatorname{blank}^{\prime\prime}

& = &

& = &

\operatorname{blank} \, \cdot\,

\operatorname{blank} \, \cdot \,

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\

^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\

\\

\\

| The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

| The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>

|}

|}

These definitions can be made a little more succinct by defining the following sorts of generic operators on strings:

These definitions can be made a little more succinct by defining the following sorts of generic operators on strings:

<ol style="list-style-type:decimal">

<ol style="list-style-type:decimal">

<li>The ''concatenation'' <math>\operatorname{Conc}^k</math> of the <math>k\!</math> strings <math>s_j, j = 1 \ldots k,\!</math> is defined recursively as follows:</li>

<li>The ''concatenation'' <math>\operatorname{Conc}_{j=1}^k</math> of the sequence of <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is defined recursively as follows:</li>

<ol style="list-style-type:lower-alpha">

<ol style="list-style-type:lower-alpha">

<li><math>\operatorname{Conc}^1_j s_j \ = \ s_1.</math></li>

<li><math>\operatorname{Conc}_{j = 1}^1 (s_j)_{j = 1}^k \ = \ s_1.</math></li>

<li><p>For <math>k > 1,\!</math></p>

<li><p>For <math>\ell > 1,\!</math></p>

     <p><math>\operatorname{Conc}^k_j s_j \ = \ (\operatorname{Conc}^{k-1}_j s_j) \, \cdot \, s_k.</math></p></li>

     <p><math>\operatorname{Conc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Conc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, s_\ell.</math></p></li>

</ol>

</ol>

<li>The ''surcatenation'' <math>\operatorname{Surc}^k</math> of the <math>k\!</math> strings <math>s_j, j = 1 \ldots k,\!</math> is defined recursively as follows:</li>

<li>The ''surcatenation'' <math>\operatorname{Surc}_{j=1}^k</math> of the sequence of <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is defined recursively as follows:</li>

<ol style="list-style-type:lower-alpha">

<ol style="list-style-type:lower-alpha">

<li><math>\operatorname{Surc}^1_j s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

<li><math>\operatorname{Surc}_{j=1}^1 (s_j)_{j = 1}^k \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>

<li><p>For <math>k > 1,\!</math></p>

<li><p>For <math>\ell > 1,\!</math></p>

     <p><math>\operatorname{Surc}^k_j s_j \ = \ (\operatorname{Surc}^{k-1}_j s_j) \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>

     <p><math>\operatorname{Surc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Surc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>

</ol></ol>

</ol></ol>