Changes

We are once again concerned with ''categories of structured items'' (COSIs) and the categories of mappings between them, indeed, the two ideas are all but inseparable, there being many good reasons to consider the very notion of structure to be most clearly defined in terms of the brands of "arrows", maps, or morphisms between items that are admitted to the category in view.

We are once again concerned with ''categories of structured items'' (COSIs) and the categories of mappings between them, indeed, the two ideas are all but inseparable, there being many good reasons to consider the very notion of structure to be most clearly defined in terms of the brands of "arrows", maps, or morphisms between items that are admitted to the category in view.

At the level of the ''primary arithmetic'' (PAR), we have a set-up like this:

At the level of the ''primary arithmetic'', we have a set-up like this:

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The object domain '''O''' is the boolean domain '''B''' = {Falsity, Truth}, the semiotic domain '''S''' is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' ''D''_en, ''D''_ex : '''S''' → '''O'''.

The object domain <math>O\!</math> is the boolean domain <math>\mathbb{B} = \{ \operatorname{falsity}, \operatorname{truth} \},</math> the semiotic domain <math>S\!</math> is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' <math>\operatorname{den}_\text{en}, \operatorname{den}_\text{ex} : S \to O.</math>

Either one of the denotation maps induces the same partition of '''S''' into REC's, a partition whose structure is suggested by the following two sets of strings:

Either one of the denotation maps induces the same partition of <math>S\!</math> into RECs, a partition whose structure is suggested by the following two sets of strings:

: {<code>" ", "(( ))", "(( )( ))", ...</code>},

: {<code>" ", "(( ))", "(( )( ))", ...</code>},