On Quantales

Haruhiko Kozawa

                         matrix@isc.chubu.ac.jp

Graduate School of Engineering, Chubu University

Introduction

Propositional system and Hilbert space

We shall call a question every experiment leading to alternative of which the terms are “yes” and “no”. When the physical system has been prepared in such a way that physicist may affirm that in the event of an experiment the result will be “yes”, we shall say that the question is true. For certain pairs of questions β, γ, one may have following property: “If the physical system is prepared in such a way that β is ture, then one is sure that γ is ture.” We shall denote it as β < γ. By definition, two questions β and γ are equivalent if one has β < γ and γ < β. We shall name a proposition, and denote by b, the equivalent class containing the quesiton β. Then, the set of propositions L is a lattice which is complete, orthocomplemented, and weakly moduar. We call it a propositional system. In propositional system L, two propositions b and c are said to be compatible if the sublattice generated by {b, b^{⊥}, c, c^{⊥}} is distributive.

Let ℙ(H) be the set of projection of Hilbert space H. If P_{1}, P_{2} ∈ ℙ(H)and K = P_{1}(H), L = P_{2}(H), then we define

1. P_{1} < P_{2} if and only if K ⊂ L

2. P_{1} ∨ P_{2} is projection of H onto K ∨ L.

3. P_{1} ∧ P_{2} is projection of H onto K ∩ L.

Then, ℙ(H) is complete lattice, and it is easy to verify the weak modularity relation. In ℙ(H), commutativity is completely equivalent to compatibility. Thus, ℙ(H) is the lattice which is which is complete, orthocomplemented, and weakly moduar.

By generalizing these results, the propositional systems can always realized by the lattice of closed subspaces of a generalized Hilbert space.

A Quantale

A quantale is a complete lattice Q with an associative binary operation & which is distribution over arbitrary joins:

a& ⋁ S = ⋁{a&s | s ∈ S} (⋁ S)&a = ⋁{s&a | s ∈ S}

Now, we consider a quantale Q(ℙ(H)) where

Q(ℙ(H)) = {φ : ℙ(H) → ℙ(H) | (⋁_{i} P_{i})φ = ⋁_{i}(P_{i})φ, ∀{P_{i}} ⊂ ℙ(H)}

and we shall show that ℙ(H) is embedded in Q(ℙ(H)). The multiplication operater & on Q(ℙ(H)) is interpreted as “ and then ”.

We review the quantales in C.J.Mulvey and J.W.Pelletier[4] in this paper.

Quantales

Definition 1 A complete lattice Q with an associative binary operation & is called a quantale if

a&(⋁ b_{i}) = ⋁(a&b_{i})
(⋁ a_{i})&b = ⋁(a_{i}&b)
∀a, b, a_{i}, b_{i} ∈ Q.

We denote the top and the bottom elements of Q by 1 and 0 respectively,

⋁ Q = 1, ⋁ ∅ = 0.

Definition 2 A quantale Q is said to be unital if there exits a unit element e ∈ Q satisfying

e&a = a = a&e ∀a ∈ Q.

Definition 3 A quantale Q with a unary operation *, ⟨Q, ∗⟩ is said to be involutive if

a^{∗∗} = a
(a&b)^{∗} = b^{∗}&a^{∗}
(⋁ a_{i})^{∗} = ⋁ a_{i}^{∗}.

Definition 4 An element a ∈ Q of quantale Q is said to be right-sided if

∀b ∈ Q(a&b ≤ a).

Similary, a ∈ Q of quantale Q is said to be left-sided if b&a ≤ a for all b ∈ Q. We denote R(Q) and L(Q), respectively the subsets of right-sided and of left-sided elements of Q.

Definition 5 A unital and involutive quantale ⟨Q, &, ∗, e⟩ is called a Gelfand quntale if

a&a^{∗}&a = a ∀a ∈ R(Q).

The right(left)-adjoint of & is denoted by →_{r} (→_{l}):

(a →_{r} b) = ⋁{x ∈ R(Q) | a&x ⩽ b}

(a →_{l} b) = ⋁{x ∈ L(Q) | x&a ⩽ b}

Then we have

a&(a →_{r} b) ⩽ b; (a →_{l} b)&a ⩽ b.

If a ∈ R(Q), then a^{∗} →_{r} 0 is R(Q) and denoted by a^{⊥}; if a ∈ L(Q), then a^{∗} →_{l} 0 is L(Q) and denoted

by ^{⊥}a.

lemma 6 ([4]) For any Gelfand quantale Q, the R(Q) of right-sided elements is a quantale, and has the following properties:

1. a&a = a 2. a ∧ b ≤ a^{∗}&b

3. a ∧ a^{⊥} = 0 for every a, b ∈ Q. Proof quantale: For any a_{i} ∈ R(Q)∀i and any c ∈ Q, then

(⋁_{i} a_{i})&c = ⋁_{i}(a_{i}&c)
≤ ⋁_{i} a_{i}.

Then, ⋁_{i} a_{i} ∈ R(Q). Similary, ⋀_{i} a_{i} ∈ R(Q). Thus, R(Q) is a complete lattice. Since R(Q) ⊆ Q, R(Q) is clearly a quantale.

a&a=a: For any a ∈ R(Q), we have a&a ≤ a = a&a^{∗}&a ≤ a&a. a∧b≤ a^{∗}&b: Since a^{∗}&a is left-sided, a = a&a^{∗}&a ≤ 1&a^{∗}&a ≤ a^{∗}&a. Thus,

a ∧ b ≤ (a ∧ b)^{∗}&(a ∧ b) ≤ a^{∗}&b.

a∧a^{⊥} =0:

a ∧ a^{⊥} ≤ a^{∗}&a^{⊥} = a^{∗}&(a^{∗} →_{r} 0) = 0

□

Theorem 7 ([4]) For some Hilbert space H, Q(ℙ(H)) is the quantale of sup-preserving mappings from sup-lattice ℙ(H) to itself.

Q(ℙ(H)) = {φ : ℙ(H) → ℙ(H) | (⋁_{i} P_{i})φ = ⋁_{i}(P_{i})φ, ∀{P_{i}} ⊂ ℙ(H)}

forms a Gelfand quantale, where

φ ≤ ψ \underleftrightarrow{def} (M)φ ≤ (M)ψ for allM ∈ ℙ(H)
(M)(⋁_{i} φ_{i}) {def \atop =} ⋁_{i}(M)φ_{i} for all M ∈ ℙ(H)
(M)φ&ψ {def \atop =} ((M)φ)ψ for all M ∈ ℙ(H)
(M)φ^{∗} {def \atop =} (⋁\{N ∈ ℙ(H) | (N)φ ≤ M^{⊥}\})^{⊥} for all M ∈ ℙ(H)
(M)e {def \atop =} M for all M ∈ ℙ(H).

Proof We define λ_{M}, for M ∈ ℙ(H) by

(N)λ_{M} = \{ \begin{matrix}0 & if N ≤ M \\ H & otherwise. \end{matrix}

It is clear that λ_{M} is a sup-preserving mapping from ℙ(H) to ℙ(H) and that

λ_{M}&φ ≤ λ_{M}

for all φ ∈ Q(ℙ(H)). Moreover, if φ is any right-sided element of Q(ℙ(H)), we must φ&1 ≤ φ where

(N)1 = \{ \begin{matrix}0 & if N = 0 \\ H & otherwise. \end{matrix}

Then, clearly φ must be of the form λ_{M}, where

M = ⋁{N ∈ ℙ(H) | (N)φ = 0}.

Thus, sup-preserving mappings of the form λ_{M} are the right-sided elemnts of Q(ℙ(H)). Similary, suppreserving mappings of the form κ_{M} are the left-sided elemnts of Q(ℙ(H)), where

(N)κ_{M} = \{ \begin{matrix}0 & if N = 0 \\ M & otherwise. \end{matrix}

Now we shall show that Q(ℙ(H)) is Gelfand quantale, that is, that

λ_{M}&λ_{M}^{∗}&λ_{M} = λ_{M}

for all M ∈ ℙ(H), turns on the fact λ_{M}^{∗} = κ_{M^{⊥}}. For, by definition

(N)λ_{M}^{∗} = ( ⋁_{(L)λ_{M} ≤N^{⊥}} L)^{⊥}.

Since (L)λ_{M} ≤ N^{⊥} when N = 0 and L is any projection on H or when N ≠ 0 and L ≤ M, we see that for N = 0

(0)λ_{M}^{∗} = H^{⊥} = 0

and for N ≠ 0

(N)λ_{M}^{∗} = M^{⊥}

which shows that

λ_{M}^{∗} = κ_{M^{⊥}}.

This completes the proof of the thorem, since it is easy to see that λ_{M}&κ_{M^{⊥}}&λ_{M} = λ_{M}. □

From this proof, the operation ^{∗} is an isomorphism from R(Q(ℙ(H))) onto L(Q(ℙ(H))).

Theorem 8 ([4]) For any Hilbert space H, the mapping

χ : ℙ(H) → R(Q(ℙ(H)))

defined by

(M)χ = λ_{M^{⊥}}

is an isomorphism of orthocomplemented sup-lattice.

Proof First, we shall show surjective. From previous theorem’s proof, if φ ∈ R(Q(ℙ(H))), then

φ = λ_{M}
M = ⋁{N ∈ ℙ(H) | (N)φ = 0}.

Since ℙ(H) is orthocomplemented sup-lattice, we have M, M^{⊥} ∈ ℙ(H). That is,

∃M^{⊥} ∈ ℙ(H)(M^{⊥})χ = λ_{M^{⊥⊥}} = λ_{M}.

Therefore, mapping χ is surjective. It is injective, because if M ≤ N and M ≠ N, then

(M^{⊥})λ_{M^{⊥}} = 0, (M^{⊥})λ_{N^{⊥}} = 1

and if M ≰ N, then

(N^{⊥})λ_{M^{⊥}} = 1, (N^{⊥})λ_{N^{⊥}} = 0.

The mapping χ preserves order, since

M ≤ N ⇒ N^{⊥} ≤ M^{⊥} ⇒ λ_{M^{⊥}} ≤ λ_{N^{⊥}} ⇒ (M)χ ≤ (N)χ.

Finally, we note that

(M^{⊥})χ = λ_{M^{⊥⊥}} = (λ_{M^{⊥}})^{⊥} = ((M)χ)^{⊥}.
□

From this theorem, it follows that ℙ(H) is extended to Gelfand quantale Q(ℙ(H)). The quantale valued logic is the logic for propositions represented by Gelfand quantale Q(ℙ(H)).

References

[1] C.Piron, Fundations of Quantum Pysics, W.A.Benjamin, Inc. 1976

[2] David W. Cohen, An Introduction to Hilbert space and Quantum Logic, Springer-Verlag New York Inc. 1989

[3] 竹内外史線形代数と量子力学裳華房 1981

[4] Christopher J. Mulvey and Joan Wick Pelletier, A Quantisation of the Calculus of Relations, Canadian Mathematical Society Conference Proceeding Volume 13, 1992 pp.345-360