Functional Monadic Bounded Algebras

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of “relatively functional” algebra, and show that every member of MBA is isomorphic to a relatively functional one. In [1], an equationally defined class MBA of monadic bounded algebras was introduced. Each of these algebras comprises a Boolean algebra B with a distinguished element E, thought of as an existence predicate, and an operator ∃ on B reflecting the properties of the existential quantifier in logic without existence assumptions. MBA was shown to be generated by a certain proper subclass FMBA of algebras isomorphic to algebras of Boolean-valued functions. In this paper we characterise FMBA as consisting precisely of those monadic bounded algebras in which ∃E = 0 or ∃E = 1. So FMBA is defined by a disjunction of two equations. We also define a weaker notion of “relativised” functional algebra and show that every monadic bounded algebra is isomorphic to one of these more general functional ones. The paper builds on [1], with which the reader is assumed to be familiar. We review the definition of FMBA. Let B be a Boolean algebra, X a set, and XE ⊆ X. The set B of all functions from X to B is a Boolean algebra with respect to the pointwise operations. A Boolean subalgebra A of B with a distinguished member E of A is called a functional monadic bounded algebra, with domain (X,XE) and distinguished function E, or more briefly a functional MBA, iff (F1) E(x) = 1 for every x ∈ XE ; (F2) for every p ∈ A, both ∨ {p(x) | x ∈ XE} and ∨ {p(x) ∧ E(x) | x ∈ X} exist in B and are equal; and (F3) for every p ∈ A, A contains the constant function ∃p on X, defined by