On the methods of mechanical non-theorems (latest version)

Definition Let U be a set. The full n-dimensional directed many-sorted polyadic set algebra over U or
the full dMsPs n over U for short, is the
U = (B(U n ), . . . , B(U n ), r U , p U , s U , c U 0 , . . . , c U n−1) of dMsP n -signature where
.
B(U n ) = (P(U n ), ∪, −, ∅) is the boolean algebra of subsets of the set U n .
r U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x n−1 , x 0 , x 1 . . . , x n−2 ) ∈ X}, i.e. subtract one from each
index modulo n,
p U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x 1 , x 0 , x 2 . . . , x n−1 ) ∈ X}, i.e. swap the first and the second
component,
s U (X) = {(x 0 , . . . , x n−1 ) ∈ U n |(x 1 , x 1 , x 2 . . . , x n−1 ) ∈ X}, i.e. overwrite the first component
with the second.
c U 0 (X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (y, x 1 , . . . , x n−1 ) ∈ X}
.
c U i (X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (x 0 , . . . , x i−1 , y, x i+1 , . . . , x n−1 ) ∈ X}
c U n−1(X) = {(x 0 , . . . , x n−1 ) ∈ U n |∃y(y ∈ U ∧ (x 0 , . . . , x n−2 , y) ∈ X}
Example The full dMsPs 3 over K. This algebra has four uncountable sorts, all equal to P(K 3 ).
The operations of this algebra are defined as in example 3.1.3. We are particularly interested in
finitely generated sub-algebras of full dMsPs n ’s over infinite sets, as these are suitable as objects of
computation as long as they are abstract, see A. Rognes [Rog09].
Definition Let A = (B n , . . . , B 0 , . . .) and A ′ = (B ′ n, . . . , B ′ 0, . . .) be two algebras of dMsP n -
signature. A dMsP n -homomorphism from A to A ′ is a tuple f = (f n , . . . , f 0 ) of mappings such
that
f n : B n → B ′ n is a boolean homomorphism,
.
f 0 : B 0 → B ′ 0 is a boolean homomorphism,
f n preserves the operations r, p and s,
for i < n it is the case that f i (c i (x)) = c ′ i(f i+1 (x)).
The homomorphism f is said to be a dMsP n -embedding if each of f 0 , . . . , f n is a boolean embedding.
Using embeddings we now define the objects which our main result is about.
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