Changes

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.

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If <math>\mathfrak{L}</math> is an arbitrary formal language over an alphabet of the sort that

If !L! is an arbitrary formal language over an alphabet of the sort that

we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.</math>

we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!,

then there are a number of basic structural relations that can be defined

on the strings of !L!.

1. z is the "concatenation" of z_1 and z_2 in !L! if and only if

 

| 1. || <math>s\!</math> is the ''concatenation'' of <math>s_1\!</math> and <math>s_2\!</math> in <math>\mathfrak{L}</math> if and only if

    z_1 is a sentence of !L!, z_2 is a sentence of !L!, and

 

| &nbsp; || <math>s_1\!</math> is a sentence of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and

    z  = z_1 · z_2.

 

| &nbsp; || <math>s = s_1 \cdot s_2.</math>

2. z is the "concatenation" of the k strings z1, ..., z_k in !L!,

 

| 2. || <math>s\!</math> is the ''concatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>

    if and only if z_j is a sentence of !L!, for all j = 1 to k, and

 

| &nbsp; || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>j = 1 \ldots k,</math> and

    z  = Conc^k_j  z_j  = z_1 · ... · z_k.

 

| &nbsp; || <math>s = \operatorname{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k.</math>

3. z is the "discatenation" of z_1 by t if and only if

 

| 3. || <math>s\!</math> is the ''discatenation'' of <math>s_1\!</math> by <math>t\!</math> if and only if

    z_1 is a sentence of !L!, t is an element of !A!, and

 

| &nbsp; || <math>s_1\!</math> is a sentence of <math>\mathfrak{L},</math> <math>t\!</math> is an element of <math>\mathfrak{A},</math> and

    z_1  = z · t.

 

| &nbsp; || <math>s_1 = s \cdot t.</math>

    When this is the case, one more commonly writes:

 

| &nbsp; || When this is the case, one more commonly writes:

    z  = z_1 · t^-1.

| &nbsp; || <math>s = s_1 \cdot t^{-1}.</math>

4.  z is a "subclause" of !L! if and only if

4.  z is a "subclause" of !L! if and only if