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Peirce algebras

Brink, Chris and Britz, Katarina and Schmidt, Renate A.

July 1992, 22 pages.

Status: available - back from printing

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.

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  AUTHOR = {Brink, Chris and Britz, Katarina and Schmidt, Renate A.},
  TITLE = {Peirce algebras},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-92-229},
  MONTH = {July},
  YEAR = {1992},
  ISSN = {0946-011X},