Changes

</ol></ol>

</ol></ol>

The definitions of these syntactic operations can now be organized in a slightly better fashion, for both conceptual and computational purposes, by making a few additional conventions and auxiliary definitions.

The definitions of these syntactic operations can now be organized in a slightly

 

better fashion, for both conceptual and computational purposes, by making a few

additional conventions and auxiliary definitions.

 

 

 

1.  The conception of the k-place concatenation operation

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

    can be extended to include its natural "prequel":

    Conc^0  =  ""  =  the empty string.

<p>The ''precatenation'' <math>\operatorname{Prec} (s_1, s_2)</math> of the two strings <math>s_1, s_2\!</math> is the string that is defined as follows:</p>

    Next, the construction of the k-place concatenation can be

<p><math>\operatorname{Prec} (s_1, s_2) \ = \ s_1 \cdot s_2.</math></p></li>

    a.  The "precatenation" Prec(z_1, z_2) of the two strings

        z_1, z_2 is the string that is defined as follows:

<p>The ''concatenation'' of the sequence of <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated precatenation over the sequence of <math>k + 1\!</math> strings that begins with the string <math>s_0 = \operatorname{Conc}^0 \, = \, ^{\backprime\backprime\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</p></li>

        Prec(z_1, z_2)  = z_1 · z_2.

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

<p><math>\operatorname{Conc}^0_j s_j \ = \ \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}.</math></p></li>

        strings that begins with the string z_0 = Conc^0 = "" and then

        i.  Conc^0_j  z_j  =  Conc^0 =  "".

<p>For <math>k > 0,\!</math></p>

        ii. For k > 0,

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

2.  The conception of the k-place surcatenation operation

2.  The conception of the k-place surcatenation operation

     can be extended to include its natural "prequel":

     can be extended to include its natural "prequel":