Quantale valued set theory

Satoko Titani

                           titani@chubu.ac.jp

Chubu University

Introduction

Observation of a physical system is described by using a Hilbert space. States of the physical system are represented as vectors of the Hilbert space H, and observables (physical quantities) are represented as self-adjoint operators on H. A proposition in quantum theory, which is considered as a property of states, are represented as a projection p on H such that

“The proposition is true in state x” is certain if and only if x is in the range of p

Projections on a Hilbert space H form a complete orthomodular lattice with respect to the inclusion of their ranges. We denote the complete orthomodular lattice by ℙ(H).

In [4] and [6], G.Takeuti developed the quantum set theory on the ℙ(H)-valued universe V^{ℙ(H)}.

V_{α}^{ℙ(H)} = {u | ∃β <α ∃Du⊂V_{β}^{ℙ(H)}(u : Du → ℙ(H))}
V^{ℙ(H)} = ⋃_{α∈On} V_{α}^{ℙ(H)}

In his quantum set theory, he defined implication →_{q} by

φ →_{q} ψ \underleftrightarrow{def} φ^{⊥} ∨ (φ ∧ ψ),

and defined the interpretation of equality =_{q} by

[[u =_{q} v]] = ⋀_{x∈Du}(u(x) →_{q} [[x ∈_{q} v]]) ∧ ⋀_{x∈Dv}(v(x) →_{q} [[x ∈_{q} u]])
[[u ∈_{q} v]] = ⋁_{x∈Dv}[[u =_{q} x]] ∧ v(x).

The operation →_{q} is an implication in the sense that

[[φ ∧ (φ →_{q} ψ)]] ⩽ [[ψ]].

But, unfortunately,

[[φ ∧ ψ]] ⩽ [[ξ]] does not imply [[φ]] ⩽ [[ψ →_{q} ξ]],

because of non-distributivity of ℙ(H). It follows that the transitivity of =_{q} is not valid in the quantum

set theory on V^{ℙ(H)}.

The aim of this paper is to review the concepts of implication and equality in the universe V^{ℙ(H)}. We extend the orthomodular lattice ℙ(H) to a quantale. Quantale is a complete lattice Q with associative binary operation & which is distributive over arbitrary joins (Definition 1). Then we introduce the rightadjoint →_{r} of & as an implication on Q. The equality =_{r} corresponding to & satisfies the transitivity. The quantale Q is defined as an algebra consisting of sup-preserving mappings φ : p ↦ (p)φ from ℙ(H) to itself.

Q = {φ : ℙ(H) → ℙ(H) | (⋁ P)φ = ⋁{(p)φ | p ∈ P } for arbitrary P ⊂ ℙ(H)}.

Q is a Gelfand quantale with respect to composition & of operations, and ℙ(H) is embedded in Q as the subquantale R(Q) of right-sided elements (Theorem 1),

ℙ(H) ≅ R(Q) = {φ ∈ Q | φ&1 ⩽ φ} ⊂ Q,

The operator & on Q is interpreted as “and then”.

In [7] we formulated a lattice valued logic and a lattice valued set theory, by introducing a strong implication, which represents the lattice order ⩽. The lattice valued logic is the logical counterpart of general complete lattices. The completeness of the lattice valued logic was proved by Takano [5]. For an arbitrary complete lattice L, we can construct a L-valued universe V^{L}. The lattice valued set theory on the universe V^{L} was formulated as a set theory based on the lattice valued logic.

Now we take Q as the complete lattice L, and consider the quantale valued set theory V^{Q}. Q-valued logic is obtained from a lattice valued logic by adding new logical operations such as &, ^{∗}, →_{r}, →_{l}, and new inference rules for them.

Since the orthocomplemented sup-lattice ℙ(H) can be embedded in Q, V^{ℙ(H)} is considered as a subuniverse of V^{Q}. Similarly, ℙ(H) can be embedded in Q as the subquantale L(Q) of left-sided elements,

L(Q) = {φ ∈ Q | 1&φ ⩽ φ}.

This means that there are two isomorphic subuniverses V^{R(Q)} and V^{L(Q)} of V^{Q}, which are mutually conjugate.

In section 1 we briefly review the quantales in C.J.Mulvey and J.W.Pelletier [2], and then, in section 2, discuss the quantale valued set theory on V^{Q}.

1 Quantales

Definition 1 A sup-lattice (complete lattice) Q with an associative binary operation & is called a quantale, if the operation & is distributive over arbitrary joins:

a& ⋁ S = ⋁{a&s | s∈S}, (⋁ S)&a = ⋁{s&a | s∈S} for S ⊂ 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 exists a unit element e in Q such that

e&a = a = a&e for all a ∈ Q

Definition 3 An element a ∈ Q of a quantale Q is said to be right-sided (respectively, left-sided) if

a&1 ⩽ a (respectively, 1&a ⩽ a).

The set of right-sided elements (respectively, left-sided elements) of Q is denoted by R(Q) (respectively, L(Q)).

R(Q) {def \atop =} \{a∈Q | a&1 ⩽ a\}, L(Q) {def \atop =} \{a∈Q | 1&a ⩽ a\}

R(Q) and L(Q) are subquantale of Q. The quantale Q is said to be right-sided (respectively, left-sided), if every a ∈ Q is right-sided (respectively, left-sided).

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

(a →_{r} b) {def \atop =} ⋁\{x ∈ R(Q) | a&x ⩽ b\} ; (a →_{l} b) {def \atop =} ⋁\{x ∈ L(Q) | x&a ⩽ b\}.

Definition 4 Unary operation ^{∗} on a quantale Q is called an involution if 1. a^{∗∗} = a

2. (a&b)^{∗} = b^{∗}&a^{∗}

3. (⋁_{α} a_{α})^{∗} = ⋁_{α} a_{α}^{∗}. A quantale with an involution ^{∗} is called involutive quantale. If a ∈ R(Q), then (a^{∗} →_{r} 0) is an element R(Q) and denoted by a^{⊥} .

Definition 5 A unital and involutive quantale Q = (Q, ⋁, &,^{∗}, e) is called Gelfand quantale, if

a&a^{∗}&a = a for all a∈R(Q)

Lemma 1.1 ([2]) For any Gelfand quantale Q, the set R(Q) of right-sided elements is a Gelfand quantale with respect to the operations of Q, and has the following properties:

a&a = a,  a ∧ b ⩽ a^{∗}&b,  a ∧ a^{⊥} = 0  for every a, b ∈ R(Q).

Let L be an orthocomplemented sup-lattice, that is, L = (L, ⋁,^{⊥}) is a sup-lattice with operation ^{⊥} such that, for all x ∈ L and {x_{α}} ⊂ L,

x^{⊥⊥} = x,  (⋁_{α} x_{α})^{⊥} = ⋀_{α} x_{α}^{⊥},  x ∨ x^{⊥} = 1, x ∧ x^{⊥} = 0.

Then the set Q(L) of sup-preserving mappings from L to itself, φ : x ↦ (x)φ,

Q(L) = {φ:L→L | (⋁_{α} x_{α})φ = ⋁_{α}(x_{α})φ for all {x_{α}} ⊂ L},

forms a Gelfand quantal with respect to ⩽, ⋁, &, ^{∗}, e defined by

φ ⩽ ψ \underleftrightarrow{def} (x)φ ⩽ (x)ψ for all x∈L
(x)(⋁_{α} φ_{α}) {def \atop =} ⋁_{α}(x)φ_{α} for all x∈L
(x)(φ&ψ) {def \atop =} ((x)φ)ψ for all x∈L
(x)φ^{∗} {def \atop =} (⋁\{t∈L | (t)φ ⩽ x^{⊥}\})^{⊥} for all x∈L
(x)e {def \atop =} x for all x∈L

Theorem 1 ([2]) An orthocomplemented sup-lattice L is embedded in the Gelfand quantale Q(L) consisting of all sup-preserving mappings from L to itself :

L ≅ R(Q(L)) ⊂ Q(L).

Now, as L in the above Theorem 1, we take the orthocomplemented sup-lattice ℙ(H) consisting of projections on a Hilbert space H. The lattice ℙ(H) is extended to Gelfand quantale Q(ℙ(H)) by the Theorem 1. We denote the Gelfand quantale Q(ℙ(H)) simply by Q, and consider a quantale valued set theory on the Q-valued universe. Truth values of propositions in quantale valued set theory are considered as elements of Gelfand quantale Q.

On Q, we have three implications, →_{r}, →_{l}, and the basic implication → of lattice valued logic ;

a → b = \begin{cases}1 & if a ⩽ b \\ 0 & otherwise \end{cases} .

For the right-implication →_{r}, and for a, b, c ∈ R(Q), we have :

(1) a&(a →_{r} b) ⩽ b and (2) a&b ⩽ c implies b ⩽ (a →_{r} c)

→_{l} has the symmetric property.

2 Quantale valued set theory

2.1 Language of quantale valued set theory

Primitive symbols are :

1. individual variables x, y, z, ⋯

2. propositional constant E

3. predicate constants =, ∈,

4. logical symbols ∧, ∨, ¬, →, ∀, ∃, &, ^{∗}, →_{r}, →_{l}.

5. parentheses (, )

Propositional constant E and expressions of the form t_{1} = t_{2} or t_{1} ∈t_{2} with individual variables t_{1}, t_{2} are atomic formulas. Formulas are constructed from atomic formulas, by using the logical symbols. To denote formulas, we use φ, ψ, ξ, ⋯, φ(x), · · ·.

2.2 Quantale valued universe V^{Q}

Let Q = (Q, ⋁, &,^{∗} , e) be a Gelfand quantale. Q-valued universe V^{Q} is constructed by induction.

V_{α}^{Q} = {u | ∃β <α ∃Du⊂V_{β}^{Q}(u : Du → Q)}
V^{Q} = ⋃_{α∈On} V_{α}^{Q}

The least ordinal α such that u∈V_{α}^{Q} is called the rank of u, and the truth value of a formula φ is denoted by [[φ]]. u = v and u∈v for u, v ∈V^{Q} are interpreted by induction on rank of u, v.

[[u=v]] = ⋀_{x∈Du}(u(x) → [[x∈v]]) ∧ ⋀_{x∈Dv}(v(x) → [[x∈u]])
[[u∈v]] = ⋁_{x∈Dv}[[u=x]] ∧ v(x)

E is interpreted as the unit e of Q, [[E]] = e. Logical operations ∧, ∨, &, →, ∀x, ∃x, involution ^{∗}, and right- and left-implications →_{r}, →_{l} are interpreted as the corresponding operators on Q.

[[φ_{1} ∧ φ_{2}]] = [[φ_{1}]] ∧ [[φ_{2}]] ; [[φ_{1} ∨ φ_{2}]] = [[φ_{1}]] ∨ [[φ_{2}]] ; [[φ_{1} → φ_{2}]] = [[φ_{1}]] → [[φ_{2}]] [[∀xφ(x)]] = ⋀_{u∈V^{Q}}[[φ(u)]] ; [[∃xφ(x)]] = ⋁_{u∈V^{Q}}[[φ(u)]] [[φ_{1}&φ_{2}]] = [[φ_{1}]]&[[φ_{2}]] ; [[φ^{∗}]] = [[φ]]^{∗}

[[φ →_{r} ψ]] = [[φ]] →_{r} [[ψ]] ; [[φ →_{l} ψ]] = [[φ]] →_{l} [[ψ]]

In the lattice valued set theory, [[u=v]] is 1 or 0, and hence, [[u=v]] ∧ ⋁_{k} b_{k} = ⋁_{k}[[u=v]] ∧ b_{k} for u, v ∈V^{Q} and {b_{k}}_{k} ⊂ Q. It follows that equality axioms holds. Since Q is a complete lattice, all axioms of lattice valued set theory are valid on V^{Q}.

For the right-implications →_{r}, the corresponding equality =_{r} and membership relation ∈_{r} are defined by:

u =_{r} v \underleftrightarrow{def} ∀x(x∈u →_{r} x ∈_{r} v) ∧ ∀x(x∈v →_{r} x ∈_{r} u)
u ∈_{r} v \underleftrightarrow{def} ∃x((x =_{r} u)&x∈v)

Lemma 2.1 If u, v ∈ V^{R(Q)}, then [[u =_{r} v]] and [[u ∈_{r} v]] are elements of R(Q).

Theorem 2 (Equality axioms) If u, v, w ∈ V^{R(Q)}, then

[[u =_{r} v]]&[[v =_{r} w]] ⩽ [[u =_{r} w]]
[[u =_{r} v]]&[[v ∈_{r} w]] ⩽ [[u ∈_{r} w]]
[[u ∈_{r} v]]&[[v =_{r} w]] ⩽ [[u ∈_{r} w]]

Definition 6 For u ∈ V^{Q}, we define the restriction u|p ∈ V^{Q} of u to p by

D(u|p) = Du, (u|p)(x) = [[x∈u]]&p

Theorem 3 If p is a right-sided element of Q, i.e. p ∈ R(Q), and u, v ∈ V^{R(Q)}, then p ⩽ [[u =_{r} v]] if and only if [[u|p = v|p]] = 1

Proof By using Theorem 2 and distributiveity of & over ⋁.

References

[1] R.V.Kadison and J.R.Ringrose, Fundamentals of Theory of Operator Algebras, AMS 1997

[2] C.J.Mulvey and J.Wick Pelletier, Quantization of the Calculus of Relations, Canadian Mathematical Society, Conference Proceeding, Vol. 13, 1992.

[3] C.Piron, Foundations of Quantum Physics, W.A. Benjamin, Inc. 1976

[4] G.Takeuti, Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton (1978).

[5] Mitio Takano, Strong Completeness of Lattice Valued Logic

[6] G.Takeuti, Quantum Set Theory, Current Issues in Quantum Logic, eds. E.Beltrametti and B.C.van Frassen, Plenum,New York (1981) pp.303-322

[7] S.Titani, Lattice Valued Set Theory, Arch. Math. Logic 38-6(1999) pp.395-421