In this Note I prove a representation theorem for Peirce algebras
Brink, Britz and Schmidt (1994).
I show that the class of full Peirce algebras is characterised by
the class of complete and atomic Peirce algebras in which the set of
relational atoms is restricted by two conditions.
One requires simplicity.
The other requires that each relational element can be uniquely expressed
in terms of Boolean atoms, or equivalently, that each relational element
can be uniquely expressed in terms of relational
identity atoms or relational points.
The result parallels the representation theorems of
Jònsson and Tarski (1952), McKinsey (1940) and
Schmidt and Ströhlein (1985) for full relation algebras.
To this end I investigate the interrelationship of identity elements and
right ideal elements inside any relation algebra and the
interrelationship of identity elements, right ideal elements and Boolean
set elements inside Peirce algebras.
Each form a Boolean algebra.
Inside relation algebras the Boolean algebra of elements below the
identity and the Boolean algebra of right ideal elements are isomorphic.
Inside Peirce algebras each is isomorphic to the underlying Boolean
algebra which is separate from the underlying relation algebra.
As a special case, I correlate the atom sets of these three different but
isomorphic Boolean algebras.
Peirce algebras have many applications, in modal logics, in particular, dynamic logic, arrow logic, dynamic modal logic, in logics of programs and in KLONE-based knowledge representation. See Brink etal. (1994). Of particular interest is the class of concrete Peirce algebras and the class of full Peirce algebras. The first characterisation of full Peirce algebras appears in the PhD Thesis of De Rijke (1993). This is also the first published representation theorem for Peirce algebras. In his characterisation de Rijke uses two conditions (besides simplicity), one on the Boolean set algebra and another on the relation algebra. These are the algebraic analogues of two irreflexivity rules required for the completeness proof of the logical analogue of Peirce algebras, dynamic modal logic. This Note shows that in Peirce algebras, because the Boolean set algebra is determined by the relation algebra, one condition, namely one on the relation algebra, is sufficient for the representation theorem. In contrast to the proof of de Rijke which is obtained from the completeness proof of dynamic modal logic, the proof we present is algebraic.