Changes

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The definitions of these syntactic operations can now be organized in a slightly better fashion by making a few additional conventions and auxiliary definitions.

The definitions of these syntactic operations can now be organized in a slightly better fashion by making a few additional conventions and auxiliary definitions.

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

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

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

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<p><math>\operatorname{Surc}^0_j s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p></li>

<p><math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p></li>

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

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

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

<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Subc}(\operatorname{Surc}_{j=0}^{\ell - 1} s_j, s_\ell).</math></p></li>

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Notice that the expressions <math>\operatorname{Conc}^0_j s_j</math> and <math>\operatorname{Surc}^0_j s_j</math> are defined in such a way that the respective operators

Notice that the expressions <math>\operatorname{Conc}_{j=0}^0 s_j</math> and <math>\operatorname{Surc}_{j=0}^0 s_j</math> are defined in such a way that the respective operators <math>\operatorname{Conc}^0</math> and <math>\operatorname{Surc}^0</math> simply ignore, in the manner of constants, whatever sequences of strings <math>s_j\!</math> may be listed as their ostensible arguments.

<math>\operatorname{Conc}^0</math> and <math>\operatorname{Surc}^0</math> basically "ignore", in the manner of constants, whatever sequences of strings <math>s_j\!</math> may be listed as their ostensible arguments.

Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.