Semiotics to Morphogrammatics

Anybody who can identify a sign, say "a", would accept that such an identification is not insisting on the small differences between different occurrences of the sign or letter "a". It would be ridicules to say that a letter "a" in red ink has not the same alphabetic meaning, i.e., to be the letter "a" if written, in the same way and having the same form, with black ink. A letter "a" is a letter "a" independent of physical differences, at least as long as the letter can be identified as the letter "a". In other words, a letter can be identified as such a letter only if it can be strictly separated from its environment. If the environment is disturbing too much the occurrence of the letter it can not be clearly identified. This interplay between identification and separation is well known in semiotics and has practical relevence for OCR software.
Given two letters "a" and "b", strings can be produced by concatenation, "aa", "bb" and "ab", "ba".
Now we learnt before that an "a" is an "a" and thus a "b" is a "b". Obviously, "aa" and "bb" are different, but also "ab" and "ba". And this is working for all sets of letters we can identify.
This ability of identification has a very old tradition. It is independent from specific languages, natural or artificial.
But slowly it gets quite boring!
Our childern are fit in it and our computers are succeeding well.
On the other hand, more or less all our scientific and especially our mathematics is based on sign systems.

Why should we make such a big thing to separate, say, "aa" from "bb", and "ab" from "ba"?
Are they not the same? There is no interesting difference between "aa" and "bb" and the same for "ab" and "ba". To insist that "aa" is different from "bb" is not less annoying than to insist that a green letter "a" is alphabetically different from a black letter "a".

Just for fun we could accept such a move away from the letter game of our childhood and academics. From now on we are interested only in patterns of letters and not in letters any more. We could call this move a pattern-oriented approach to scripts, or even, to be scientifically trendy, a morphic abstraction. Morphe in Greek means form, pattern or better, Gestalt. And such inscriptions of patterns can be called morphograms.
Such a game would be useless if it wouldn’t produce new rules. So, what are the new rules of the game?
To answer this question, we remember the rules of the games of letters. Letters, marks, signs, characters comes as atomic signs and can be connected to compound signs. The atomic signs are collected in a signs repertoire, also called alphabet. It is presumed that the numbers of signs of an alphabet can be finite or even infinite.The compound signs are then produced out of such an alphabet with the help of rules. The basic rule is the rule defined by concatenation. As usual, there is also a dual approach. Instead of concatenation we can chose its dual, substitution. Such compound signs are called words. Both together, the alphabet and the rules, are producing a word algebra. The algebra determines the properties of the rules.

Monads
"Words" of length 1 are called in a morphogrammatic game, monadic words, or monads. We can think of a plurality of monads, like (a), (b) or (c). But if we bring those isolated monads together, we discover that they are all the same, i.e., monads. They are involved in a morphogrammmatic equivalence.
On the semiotic side, we see, that all different atomic signs are not equal but different. Later, we can introduce a less "semiotic", i.e., sign-focussed, approach to morphograms and will be able to avoid such a paradox wording of the sameness of a plurality of monads. In fact, there is, morphogrammatically, only one monad. This fact doesn’t make a monad "holy", in the sense of Pythagoreaism.
We can also bring two monads together, to form a coalition or being concatenated. But instead of being chained, monads have only the chance to cooperate as the same or as a different to the existing monad or, later, morphogram.

Thus, (a), (a) ––> {(aa), (ab)}
or (a), (b) ––> {(aa), (ab)}
and (ab), (a) ––> {(aba), (abb), (abc)}.

The semiotic approach is still too much focussed on the objects of the game instead of the operativity of the rules (morphisms) of morphogrammatics.
Similar to the duality in category theory of objects and morphisms.

Convention
To inscribe with signs (letters, characters, marks, numbers, etc.) patterns we have to agree to a convention, say, we take (a) as the notation of a morphogram of length 1. All other representations like (b), (c), etc. are morphogrammatically the same.

This convention is not more obscure than to agree to a standard representation of a sign, say a. Remember, this sign "a" can have many occurrences.
For that, the discipline of semiotics is distincting between type and token of a sign. Tokens are inscribed on paper, types are recognized in the mind of a reader.
Types, thus, are abstractions from tokens.

Chiasm of types and tokens
Morphograms as double abstractions
Graphemic abstraction from token to type:

{a, a, a, a, a, a}/graph = {a}

Morphogrammatic abstraction from type to morphogram:

{a, b, c, d, e, f}/morph = (a)


But: conc { (aba), (a)} = {(abaa), (abab), (abac)}

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