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We prove that the Weihrauch lattice can be transformed into a
Brouwer algebra by the consecutive application of two closure operators in
the appropriate order: rst completion and then parallelization. The closure
operator of completion is a new closure operator that we introduce. It transforms
any problem into a total problem on the completion of the respective
types, where we allow any value outside of the original domain of the problem.
This closure operator is of interest by itself, as it generates a total version
of Weihrauch reducibility that is dened like the usual version of Weihrauch
reducibility, but in terms of total realizers. From a logical perspective completion
can be seen as a way to make problems independent of their premises.
Alongside with the completion operator and total Weihrauch reducibility we
need to study precomplete representations that are required to describe these
concepts. In order to show that the parallelized total Weihrauch lattice forms
a Brouwer algebra, we introduce a new multiplicative version of an implication.
While the parallelized total Weihrauch lattice forms a Brouwer algebra with
this implication, the total Weihrauch lattice fails to be a model of intuitionistic
linear logic in two dierent ways. In order to pinpoint the algebraic reasons
for this failure, we introduce the concept of a Weihrauch algebra that allows
us to formulate the failure in precise and neat terms. Finally, we show that
the Medvedev Brouwer algebra can be embedded into our Brouwer algebra,
which also implies that the theory of our Brouwer algebra is Jankov logic
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