2. ALGEBRAIC SEMIOTICS

Communication is mediated by signs, which occur in structured systems.

Semiotic systems are axiomatic theories for sign systems, including

1. sorts for signs,
2. constructors for signs, and
3. measures of their importance.

Context, including the settings of signs, is as important for meaning as signs.

For example, ``Yes'' can mean almost anything, given appropriate context.

Algebraic semiotics treats aspects of context with constructors that put signs in larger signs.

But interpretation is still needed for meaning in human sense.

Also, algebraic semiotics should be used flexibly, like music notation in performance.

Design often views some signs as representing other signs,

and asks what makes some representations better than others.

Semiotic morphisms are maps of semiotic spaces that preserve significant properties.

• Since semiotic systems are theories, not models, morphisms translate the languages of semiotic systems, not just concrete signs.
• The quality of representations is measured by how well features of the source space are preserved.
• Representations can be combined, and conceptual blending (as in cognitive linguistics) is a special case.
• Algebraic laws have been proved about combining representations.
• Case studies include web displays for proofs.

E.g., a book table of contents preserves structure and names of parts, but not content.

Design is massaging a source space, a target space, and a morphism, to achieve quality, subject to constraints.

This applies to managing an organization, and designing a website.

Design principles include:

1. The most important subsigns should map to correspondingly important subsigns in the representation;
2. The most important axioms about signs should also be satisfied by their representations; and
3. it is better to preserve form (i.e., structure) than content, if something must be sacrificed.

These principles all have precise mathematical formulations.