Monadic Bounded Algebras
Galym Akishev and Robert Goldblatt
We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture
algebraic properties of bounded quantification. The variety of all MBA's is shown to be
generated by certain algebras of two-valued propositional functions that correspond to models of
monadic free logic with an existence predicate. Every MBA is a subdirect product of such
functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness
for monadic free logic. The analysis involves the representation of MBA's as powerset algebras
of certain directed graphs with a set of ``marked'' points.
It is shown that there are only countably many varieties of MBA's, all are generated by their
finite members, and all have finite equational axiomatisations classifying them into fourteen kinds
of variety. The universal theory of each variety is decidable.
Finitely generated MBA's are shown to be finite, with the free algebra on r generators
having exactly
2 3. 2r . 2 2r - 1
elements. An explicit procedure is given for constructing
this freely generated algebra as the powerset algebra of a certain marked graph determined by
the number r.