A study of Canonicity for Bi-Implicative Algebras
Lisa Maree Fulford

Abstract:

This thesis is an analytical study of canonicity for logics with a
language consisting of constants and implications. More specifically,
logics associated with certain distinguished sub-quasivarieties of
(bi-)implicative algebras, the best known of which are the varieties
of (bi-)Hilbert algebras and (bi-)Tarski algebras. The axiomatization
of bi-implicative algebras essentially says that the binary relation 
≤ defined on A equivalently as:
                    a→b=⊤  iff  a≤b  iff  a←b=⊥
is a partial order and (A, ≤, ⊤, ⊥) is a bounded poset.

This is a case study for a program which proposes bringing inspiration
from the field of abstract algebraic logic (AAL) to the theory of
canonical extensions in the form of suggesting logical filters, or
S-filters for the appropriate associated logic S, as the "right"
choice when no all-purpose choice of filters and ideals is
available. We have chosen to move away from the lattice setting, where
the logical and the mathematically optimal choices overlap, to be able
to clearly see the power and limitations of a logic-based choice.  The
subtraction operator is added to give a clear notion of logical ideal.

Although the AAL perspective was applied in the setting of canonical
extensions of Hilbert algebras in [17], the constuction took a detour
before applying the parametrised method of canonical extensions for
posets (of [16]) with F and I taken as the down-directed upsets and
up-directed downsets respectively. This gives a canonical extension
which is not symmetric in its relationship to the logical filters and
ideals; moreover, it is an ad-hoc construction which really utilises
the powerful axiomatization of Hilbert algebras and is not applicable
in more general contexts.

Our goal is to apply the parametrised method directly, with logically
motivated choices of F and I . To define the (F, I)-extension on a
bi-implicative algebra, the algebra must also satisfy three additional
conditions (and their duals). The first, (Det), (which relates
logically to the detachment theorem) is needed to ensure that the
algebra can be embedded in its extension, and other two, (OR1) and
(OP2), (which express that → is order reversing in the first
coordinate and order preserving in the second respectively) allow → 
to be extended. The class of algebras which satisfy these axioms,
however, is still a proper extension of the the class of bi-Hilbert
algebras. Furthermore, this generalised setting allows us to study
canonicity in a more modular way than, for example in [17]. Amongst
our results we show canonicity of the bi-implicative axioms in the
presense of the conditions ⊤←a=⊤and a→⊥=⊥ where intuitively, the
first expresses that all non-bottom states are consistent and the
second that all non-top states are

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