Changes

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:

In order to systematize the relations that strictures and straits placed

at higher levels of complexity, constraint, information, and organization

have with those that are placed at the associated lower levels, I introduce

the following pair of definitions:

The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded

The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.

within a frame of discussion where the number of places is limited to k,

is the stricture of the form "X x ... x S_j x ... x X".  In the proper

context, this can be written more succinctly as the stricture "S_j_<j>",

an assertion that places the j^th set in the j^th place of the product.

The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained

The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained

to a frame of discussion where the number of places is restricted to k,

to a frame of discussion where the number of places is restricted to k,