GoI for MELL: the *-autonomous structure
Recall that when u and v are p-isometries we say they are dual when uv is nilpotent, and that denotes the set of nilpotent operators. A type is a subset of that is equal to its bidual. In particular is a type for any . We say that X generates the type .
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The tensor and the linear application
If u and v are two p-isometries summing them doesn't in general produces a p-isometry. However as pup^{ * } and qvq^{ * } have disjoint domains and disjoint codomains it is true that pup^{ * } + qvq^{ * } is a p-isometry. Given two types A and B, we thus define their tensor by:
Note the closure by bidual to make sure that we obtain a type.
From what precedes we see that is generated by the internalizations of operators on of the form:
Remark: This so-called tensor resembles a sum rather than a product. We will stick to this terminology though because it defines the interpretation of the tensor connective of linear logic.
The linear implication is derived from the tensor by duality: given two types A and B the type is defined by:
- .
Unfolding this definition we get:
- .
The identity
Given a type A we are to find an operator ι in type , thus satisfying:
- .
An easy solution is to take ι = pq^{ * } + qp^{ * }. In this way we get ι(pup^{ * } + qvq^{ * }) = qup^{ * } + pvq^{ * }. Therefore (ι(pup^{ * } + qvq^{ * }))^{2} = quvq^{ * } + pvup^{ * }, from which one deduces that this operator is nilpotent iff uv is nilpotent. It is the case since u is in A and v in .
It is interesting to note that the ι thus defined is actually the internalization of the operator on given by the matrix:
- .
We will see once the composition is defined that the ι operator is the interpretation of the identity proof, as expected.
The execution formula, version 1: application
Definition
Let u and v be two operators; as above denote by u_{ij} the external components of u. If u_{11}v is nilpotent we define the application of u to v by:
App(u,v) = u_{22} + u_{21}v | ∑ | (u_{11}v)^{k}u_{12} |
k |
.
Note that the hypothesis that u_{11}v is nilpotent entails that the sum
∑ | (u_{11}v)^{k} |
k |
is actually finite. It would be enough to assume that this sum converges. For simplicity we stick to the nilpotency condition, but we should mention that weak nilpotency would do as well.
Theorem
If u and v are p-isometries such that u_{11}v is nilpotent, then App(u,v) is also a p-isometry.
Proof. Let us note E_{k} = u_{21}v(u_{11}v)^{k}u_{12}. Recall that u_{22} and u_{12} being external components of the p-isometry u, they have disjoint domains. Thus it is also the case of u_{22} and E_{k}. Similarly u_{22} and E_{k} have disjoint codomains because u_{22} and u_{21} have disjoint codomains.
Let now k and l be two integers such that k > l and let us compute for example the intersection of the codomains of E_{k} and E_{l}:
As k > l we may write . Let us note so that . We have:
because u_{11} and u_{12} have disjoint codomains, thus .
Similarly we can show that E_{k} and E_{l} have disjoint domains. Therefore we have proved that all terms of the sum App(u,v) have disjoint domains and disjoint codomains. Consequently App(u,v) is a p-isometry.
Theorem
Let A and B be two types and u a p-isometry. Then the two following conditions are equivalent:
- ;
- for any we have:
- u_{11}v is nilpotent and
- .
Proof. Let v and w be two p-isometries. If we compute
we get a finite sum of monomial operators of the form:
- ,
- or
- ,
for all tuples of (nonnegative) integers such that .
Each of these monomial is a p-isometry. Furthermore they have disjoint domains and disjoint codomains because their sum is the p-isometry (u.(pvp^{ * } + qwq^{ * }))^{n}. This entails that (u.(pvp^{ * } + qwq^{ * }))^{n} = 0 iff all these monomials are null.
Suppose u_{11}v is nilpotent and consider:
- .
Developping we get a finite sum of monomials of the form:
- 5.
for all tuples such that and k_{i} is less than the degree of nilpotency of u_{11}v for all i.
Again as these monomials are p-isometries and their sum is the p-isometry (App(u,v)w)^{n}, they have pairwise disjoint domains and pairwise disjoint codomains. Note that each of these monomial is equal to q^{ * }Mq where M is a monomial of type 4 above.
As before we thus have that iff all monomials of type 5 are null.
Suppose now that and . Then, since (0 belongs to any type) u.(pvp^{ * }) = pu_{11}vp^{ * } is nilpotent, thus u_{11}v is nilpotent.
Suppose further that . Then u.(pvp^{ * } + qwq^{ * }) is nilpotent, thus there is a N such that (u.(pvp^{ * } + qwq^{ * }))^{n} = 0 for any . This entails that all monomials of type 1 to 4 are null. Therefore all monomials appearing in the developpment of (App(u,v)w)^{N} are null which proves that App(u,v)w is nilpotent. Thus .
Conversely suppose for any and , u_{11}v and App(u,v)w are nilpotent. Let P and N be their respective degrees of nilpotency and put n = N(P + 1) + N. Then we claim that all monomials of type 1 to 4 appearing in the development of (u.(pvp^{ * } + qwq^{ * }))^{n} are null.
Consider for example a monomial of type 1:
with . Note that m must be even.
If for some then thus our monomial is null. Otherwise if i_{2k} < P for all k we have:
thus:
- .
Now if then . Otherwise thus
- .
Since N is the degree of nilpotency of App(u,v)w we have that the monomial:
is null, thus also the monomial of type 1 we started with.
Corollary
If A and B are types then we have:
- .
As an example if we compute the application of the interpretation of the identity ι in type to the operator then we have:
- .
Now recall that ι = pq^{ * } + qp^{ * } so that ι_{11} = ι_{22} = 0 and ι_{12} = ι_{21} = 1 and we thus get:
- App(ι,v) = v
as expected.
The tensor rule
Let now A,A',B and B' be types and consider two operators u and u' respectively in and . We define an operator by:
Once again the notation is motivated by linear logic syntax and is contradictory with linear algebra practice since what we denote by actually is the internalization of the direct sum .
Indeed if we think of u and u' as the internalizations of the matrices:
- and
then we may write:
Thus the components of are given by:
- .
and we see that is actually the internalization of the matrix:
We are now to show that if we suppose uand u' are in types and , then is in . For this we consider v and v' respectively in A and A', so that pvp^{ * } + qv'q^{ * } is in , and we show that .
Since u and u' are in and we have that u_{11}v and u'_{11}v' are nilpotent and that App(u,v) and App(u',v') are respectively in B and B', thus:
- .
But we have:
Therefore is nilpotent. So we can compute :
thus lives in .
Other monoidal constructions
Contraposition
Let A and B be some types; we have:
Indeed, means that for any v and w in respectively A and we have which is exactly the definition of .
We will denote the operator:
where u_{ij} is given by externalization. Therefore the externalization of is:
- where is defined by .
From this we deduce that and that .
Commutativity
Let σ be the operator:
- σ = ppq^{ * }q^{ * } + pqp^{ * }q^{ * } + qpq^{ * }p^{ * } + qqp^{ * }p^{ * }.
One can check that σ is the internalization of the operator S on defined by: . In particular the components of σ are:
- σ_{11} = σ_{22} = 0;
- σ_{12} = σ_{21} = pq^{ * } + qp^{ * }.
Let A and B be types and u and v be operators in A and B. Then pup^{ * } + qvq^{ * } is in and as σ_{11}.(pup^{ * } + qvq^{ * }) = 0 we may compute:
But , thus we have shown that:
- .
Distributivity
We get distributivity by considering the operator:
- δ = ppp^{ * }p^{ * }q^{ * } + pqpq^{ * }p^{ * }q^{ * } + pqqq^{ * }q^{ * } + qppp^{ * }p^{ * } + qpqp^{ * }q^{ * }p^{ * } + qqq^{ * }q^{ * }p^{ * }
that is similarly shown to be in type for any types A, B and C.
Weak distributivity
Similarly we get weak distributivity thanks to the operators:
- δ_{1} = pppp^{ * }q^{ * } + ppqp^{ * }q^{ * }q^{ * } + pqq^{ * }q^{ * }q^{ * } + qpp^{ * }p^{ * }p^{ * } + qqpq^{ * }p^{ * }p^{ * } + qqqq^{ * }p^{ * } and
- δ_{2} = ppp^{ * }p^{ * }q^{ * } + pqpq^{ * }p^{ * }q^{ * } + pqqq^{ * }q^{ * } + qppp^{ * }p^{ * } + qpqp^{ * }q^{ * }p^{ * } + qqq^{ * }q^{ * }p^{ * }.
Given three types A, B and C then one can show that:
- δ_{1} has type and
- δ_{2} has type .
Execution formula, version 2: composition
Let A, B and C be types and u and v be operators respectively in types and .
As usual we will denote u_{ij} and v_{ij} the operators obtained by externalization of u and v, eg, u_{11} = p^{ * }up, ...
As u is in we have that ; similarly as , thus , we have . Thus u_{22}v_{11} is nilpotent.
We define the operator Comp(u,v) by:
This is well defined since u_{11}v_{22} is nilpotent. As an example let us compute the composition of u and ι in type ; recall that ι_{ij} = δ_{ij}, so we get:
- Comp(u,ι) = pu_{11}p^{ * } + pu_{12}q^{ * } + qu_{21}p^{ * } + qu_{22}q^{ * } = u
Similar computation would show that Comp(ι,v) = v (we use pp^{ * } + qq^{ * } = 1 here).
Coming back to the general case we claim that Comp(u,v) is in : let a be an operator in A. By computation we can check that:
- App(Comp(u,v),a) = App(v,App(u,a)).
Now since u is in , App(u,a) is in B and since v is in , App(v,App(u,a)) is in C.
If we now consider a type D and an operator w in then we have:
- Comp(Comp(u,v),w) = Comp(u,Comp(v,w)).
Putting together the results of this section we finally have:
Theorem
Let GoI(H) be defined by:
- objects are types, ie sets A of p-isometries satisfying: ;
- morphisms from A to B are p-isometries in type ;
- composition is given by the formula above.
Then GoI(H) is a star-autonomous category.