Dual Tableaux for Peirce Algebras

Abstract

Peirce algebras are two-sorted algebras of relations and sets. A Peirce algebra consists of a Boolean algebra where sets interact with each other, a relation algebra where relations interact with each other, and two operators that relate these two structures: a set-forming operator acting on a relation and a set, and a relation-forming operator acting on a set. Peirce algebras were first introduced in a modern form in [Bri88] and studied in [BBS94,dR99,SOH04,Hir07]. However, the history of these algebras can be traced back to the work of Charles Sanders Peirce who gave the first algebraic treatment of the algebra of relations interacting with sets. Whereas De Morgan was primarily interested in the formalization of statements within the paradigm of binary relations, Peirce considered the expressions obtained as a product of a relation and a set. Brink [Bri81] named this operation the Peirce product and axiomatized it within the framework of Boolean modules. The relation forming operator of Peirce algebras may be viewed as a cylindrification. Peirce algebras provide tools for modelling program constructors in programming languages, for natural language analysis [Böt92a,Böt92b] and for knowledge representation. In particular, they provide semantics for terminological languages [Sch91,WS92].