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Abstract:
We present a two-sorted algebra, called a {\em Peirce algebra},
of relations and sets interacting with each other.
In a Peirce algebra, sets can combine
with each other as in a Boolean algebra, relations can combine with
each other as in a relation algebra, and in addition we have both
a relation-forming operator on sets (the Peirce product of Boolean
modules) and a set-forming operator on
relations (a cylindrification operation). Two applications
of Peirce algebras are given.
The first points out that Peirce algebras provide a natural algebraic
framework for modelling certain programming constructs. The second shows
that the
so-called {\em terminological logics} arising in knowledge
representation have evolved a semantics best described as a calculus
of relations interacting with sets.
