Algebraic semiotics is a formalism for describing signs, systems of signs, and mappings between sign systems. It is useful not only for analysis—the effort to describing existing signs, with which classical semiotics is primarily concerned—but more important, for synthesis—the systematic design of new signs.
From the standpoint of algebraic semiotics, a user interface is a sign, albeit an interactive and highly complex one made up of many constituent signs individually representing concepts that were used by the designers and hopefully are salient to users. Algebraic semiotics aims to provide designers with a formal, structured way to compose, decompose, and compare possible (and actual) interfaces, and hopefully in the end, to help build better ones.
To understand how to create a good design, it is useful to know what can lead to a poor one. This insight is the motivation behind the UC San Diego Semiotic Zoo. Take a look now at the Zoo's Semiotic Introduction.
In the introduction, you may have noticed how algebraic semiotics draws on classical semiotics for the concepts of sign and sign system, though the terms are used in a more precise and specific way. This only hints at the richness of algebraic semiotic descriptions; there is more to come.
Some important things to remember from the semiotic introduction are the notions of source sign and display sign, and the inferred mapping between them. The source sign is the concept the designer had in mind, and the display sign is the way he or she chose to represent it in the interface. Algebraic semiotics draws attention to the mapping, called a semiotic morphism; from this perspective, properties of this mapping affect how easily the user can understand the underlying design and how to interpret and use the interface.
Two of the Zoo's exhibits relate to user interface design more directly than the others: Unix Command Notation Stalks a Toy Algebra and One Slightly Confused Applet. Take a look at these exhibits and their accompanying explanations.
The explanations for the exhibits should give a basic idea what an algebraic semiotic analysis looks like: considering the display signs, inferring the source signs, identifying the semiotic morphism that maps from source to display, thinking about what that morphism preserves, and especially whether it preserves the most important structure in the source space. These are simple examples, of course; in a real-life analysis (or synthesis, for that matter), it would be important to spend more time thinking about the goals and social context of the interface and its users.
The Explanation for a Slightly Confused Applet introduces another concept in algebraic semiotics: level. We said before that algebraic semiotics views interfaces as complex signs made up of other signs; another way to describe this is to say an interface is a high-level sign composed of lower-level signs such as windows and buttons.
Take some time to explore the rest of the Semiotic Zoo; although the other examples aren't user interface design examples, they are relatively simple examples of algebraic semiotics in action.
The main ideas in algebraic semiotics are precise notions of sign systems and the mappings between them. The Semiotic Zoo should give a basic grasp of these, but the document Semiotic Morphisms fleshes them out in more detail.
Take special note of one point in the introduction about algebraic semiotics: although its ideas have a precise mathematical formulation, they can be used informally as well. In user interface design it will generally be inadvisable—even infeasible—to work out the entire mathematical structure of the source and display systems, let alone the morphism between them. In practice an informal approach can yield insights without the unnecessary detail of a fully developed theory; nevertheless, an understanding of how such a theory would look is useful even in taking an informal approach. An excellent way to see how the formalism works is to see it applied to some small examples, which is what the Semiotic Morphisms document does.
To describe in more depth the precise notions in algebraic semiotics, the paper introduces several key terms. Sorts are categories of signs; for example, in a user interface, there might be a sort "button" of which all the buttons are instances. There might be subsorts of the button sort, such as mode-toggle buttons and action buttons. Signs of a certain sort are generally made up of signs at lower-level sorts; constructors specify these relationships. For example, a button might be constructed from a grey rectangle and an icon or a text label. Axioms limit such constructions. There might be an axiom that says the text label on a button is never longer than a few words.
The notion of priority is of critical importance when applying algebraic semiotics to user interface design, primarily because it comes into play when appraising the quality of a semiotic morphism. Priority is an ordering on constructors, generally specifying which signs of a given sort are more important than others. A morphism that preserves in the design space the priority in the source space is clearly better than one that does not; for example, especially urgent or important messages should be the ones that most demand attention. An interface that allows such messages to appear subtly in occluded windows is likely to engender much frustration—or worse, depending on the nature of the messages and the purpose of the system.
Our aim here is only to give a general introduction to classical and algebraic semiotics as they relate to user interface design. Far more has been written about algebraic semiotics, much of which is concerned to some extent with user interface design, since that is one of the motivating reasons for its inception.
A position paper on Information Visualization and Semiotic Morphisms by Joseph Goguen and Fox Harrell treats the topic of visual representation, which is of course an integral part of user interface design, using algebraic semiotics. The examples in section 3 all involve user interfaces: a code browser that reflects how recently code has been edited, a tool for exploring a database of movies, and the various representations file-system structure available in Macintosh's Finder. The paper also contains a brief but lucid introduction to algebraic semiotics and its application to design generally.
For the more deeply interested reader, Goguen's "An Introduction to Algebraic Semiotics, with Applications to User Interface Design" (PS, PDF) provides a thorough treatment of both the theoretical and practical aspects of algebraic semiotics. It contains detailed explanation of the mathematical formalism as well as numerous examples. It also introduces conceptual blending as a special case of semiotic morphisms.
Goguen's more recent paper, "Semiotic Morphisms, Representations and Blending for User Interface Design" (PS) contains an in-depth analysis of the familiar user-interface elements, windows and scrollbars. This paper uses the language BOBJ for formal notation, but it explains the conventions of BOBJ for readers who are unfamiliar with it.