Mathematical Notes:
The following are a listing of unpublished notes I have written on
on various mathematical subjects related to theoretical computer
science.
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Strings
and Strings-of-Strings Various notions of the intuitive concept of
a string are employed in computer science. In this note we look at one
that we have found to be especially useful. Namely the notion of a string
of length n over as a set A as a mapping s:{1,...,n}-->A. We define the
corresponding category of strings, Str, and examine its properties.
Then, in the second part of the note. we look at the special case where
the set A is itself the set of strings over some other set. We define
the category,
SStr, of strings-of-strings and (yet to come) some
of its applications in computer science.
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Notes
on Birkhoff's Theorem Birkhoff's theorem is a famous theorem of algebra
proved by Garret Birkhoff in 1935. It provides a useful characterization
of algebraic varieties, that is, of equationally specifiable classes of
algebras. These notes provide a short review of the basic definitions of
universal algebra, a proof of Birkhoff's theorem, an analysis of the proof
and some examples of classes of algebras that fail to satisfy all
the requirements of the theorem.
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More to Come!