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In my DPhil thesis, I defined what I called a context algebra as a model of meaning in natural language. The idea is to mathematically formalise the notion that meaning is determined by context. It can also be viewed as the equivalent of the syntactic monoid for fuzzy languages.

Let $L$ be a function from $A^* $ to $\mathbb{R}$ where $A^* $ is the free monoid on a set $A$. For $x \in A^* $, we define the context vector $\hat{x}$ as the function from $A^* \times A^* $ to $\mathbb{R}$ as $$\hat{x}(y,z) = L(yxz)$$ It is then easy to show that the vector space generated by elements {$\hat{x} : x \in A^* $} is an algebra over the reals given multiplication defined by $\hat{x}\cdot \hat{y} = \widehat{xy}$ (it is just necessary to show that no matter which elements of $A^*$ are used to form basis elements, the definition of multiplication is the same). The algebra is associative and has unit element $\hat{\epsilon}$ where $\epsilon$ is the empty string.

You can also define a linear functional $\phi$ on the algebra by $$\phi(f) = \sum_{x,y \in A^*} f(x,y)$$ If $\phi(\hat{\epsilon})$ is finite then the algebra becomes a non-commutative probability space with the linear functional $\phi'(f) = \phi(f)/\phi(\hat{\epsilon})$.

I have not come across anyone who is aware of previous work along these lines, nevertheless, given the breadth of knowledge here on Math Overflow, my first question is

1) Is this a new idea? Is it very similar to any existing work?

As a non mathematician (but aspiring amateur), my second question is

2) Is this of interest to mathematicians? Or is it just an obscure but fairly trivial example of existing maths?

Finally,

3) What would be required to develop this to a point where it would make an interesting paper for a maths journal? Are there any points for investigation that stand out? Is there any particular journal this might be suited to?

Thanks in advance, and I hope this question is suitable for Math Overflow.

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I'm not sure that the tag "operator algebras" is that relevant or helpful - that's nothing to do with how interesting your question is, btw –  Yemon Choi Jan 29 '10 at 21:57
3  
Oh, and +1 for thinking through your question and having well-defined and self-aware aims. –  Yemon Choi Jan 29 '10 at 21:57
    
True - I was trying to fit it into one of the recommended categories. Is there a better one or should I just remove it? Thanks. –  Daoud Jan 29 '10 at 22:11
    
One idea I've had for further investigation is attempting to identify the class of algebras for which there exists a function $L$ for each instance such that the context algebra of $L$ is isomorphic to the instance. @sigfpe - thanks for fixing the tags. –  Daoud Jan 29 '10 at 22:43
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