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The sum of observables on a σ-distributive lattice effect algebra

Jiˇ r´ ı Janda, Yongming Li

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Effect algebras

Definition (Foulis, Bennett, 1994) A partial algebra (E; ⊕, 0, 1) is called an effect algebra if 0, 1 are two distinct elements and ⊕ is a partially defined binary operation on E which satisfy the following conditions for any x, y, z ∈ E: (Ei) x ⊕ y = y ⊕ x if x ⊕ y is defined, (Eii) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) if one side is defined, (Eiii) for every x ∈ E there exists a unique y ∈ E such that x ⊕ y = 1 (we put x′ = y), (Eiv) if 1 ⊕ x is defined then x = 0.

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Effect algebras

Definition (Foulis, Bennett, 1994) A partial algebra (E; ⊕, 0, 1) is called an effect algebra if 0, 1 are two distinct elements and ⊕ is a partially defined binary operation on E which satisfy the following conditions for any x, y, z ∈ E: (Ei) x ⊕ y = y ⊕ x if x ⊕ y is defined, (Eii) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) if one side is defined, (Eiii) for every x ∈ E there exists a unique y ∈ E such that x ⊕ y = 1 (we put x′ = y), (Eiv) if 1 ⊕ x is defined then x = 0. A partial order ≤ on E can be introduced by: x ≤ y iff x ⊕ z is defined and x ⊕ z = y.

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E.

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E. An effect algebra E is a lattice (σ-lattice) effect algebra if (E, ≤) is a lattice (σ-lattice),

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E. An effect algebra E is a lattice (σ-lattice) effect algebra if (E, ≤) is a lattice (σ-lattice), a monotone σ-complete if for every chain a1 ≤ a2 ≤ . . . there exists a ∈ E such that a =

i∈N ai.

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E. An effect algebra E is a lattice (σ-lattice) effect algebra if (E, ≤) is a lattice (σ-lattice), a monotone σ-complete if for every chain a1 ≤ a2 ≤ . . . there exists a ∈ E such that a =

i∈N ai.

σ-frame effect algebra – if (E, ≤) is a σ-frame, i.e., σ-complete lattice which for countable I satisfies a ∧ (

• i∈I

ai) =

• i∈I

(a ∧ ai).

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E. An effect algebra E is a lattice (σ-lattice) effect algebra if (E, ≤) is a lattice (σ-lattice), a monotone σ-complete if for every chain a1 ≤ a2 ≤ . . . there exists a ∈ E such that a =

i∈N ai.

σ-frame effect algebra – if (E, ≤) is a σ-frame, i.e., σ-complete lattice which for countable I satisfies a ∧ (

• i∈I

ai) =

• i∈I

(a ∧ ai). Remark An effect algebra E is σ-frame if and only E is σ-coframe.

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Effect algebraic partial order

With respect to ≤, 1 is the top and 0 is the bottom element of E. An effect algebra E is a lattice (σ-lattice) effect algebra if (E, ≤) is a lattice (σ-lattice), a monotone σ-complete if for every chain a1 ≤ a2 ≤ . . . there exists a ∈ E such that a =

i∈N ai.

σ-frame effect algebra – if (E, ≤) is a σ-frame, i.e., σ-complete lattice which for countable I satisfies a ∧ (

• i∈I

ai) =

• i∈I

(a ∧ ai). Remark An effect algebra E is σ-frame if and only E is σ-coframe.

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Examples of effect algebras

Boolean algebras – a ⊕ b is defined iff a ≤ b∗ in which case a ⊕ b = a ∨ b,

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Examples of effect algebras

Boolean algebras – a ⊕ b is defined iff a ≤ b∗ in which case a ⊕ b = a ∨ b, MV-algebras – a ⊕ b is defined iff a ≤ b′ in which case a ⊕ b = a ⊞ b,

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Examples of effect algebras

Boolean algebras – a ⊕ b is defined iff a ≤ b∗ in which case a ⊕ b = a ∨ b, MV-algebras – a ⊕ b is defined iff a ≤ b′ in which case a ⊕ b = a ⊞ b, Interval effect algebras – let (G; +, ≤) be a partially ordered commutative group, a ∈ G, 0 < a. Then [0, a] ⊆ G with x ⊕ y = x + y iff x + y ≤ a is an effect algebra.

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Examples of effect algebras

Boolean algebras – a ⊕ b is defined iff a ≤ b∗ in which case a ⊕ b = a ∨ b, MV-algebras – a ⊕ b is defined iff a ≤ b′ in which case a ⊕ b = a ⊞ b, Interval effect algebras – let (G; +, ≤) be a partially ordered commutative group, a ∈ G, 0 < a. Then [0, a] ⊆ G with x ⊕ y = x + y iff x + y ≤ a is an effect algebra. E(H) := [0, I] ⊆ B(H) – an interval on bounded self-adjoint linear

• perators on a complex Hilbert space H, with the usual addition,

A ≤ B if (Ax, x) ≤ (Bx, x) for all x ∈ H.

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Observables

Definition Let E be a monotone σ-complete effect algebra. An observable is a map x : B(R) → E such that (i) x(R) = 1, (ii) if A ∩ B = ∅ then x(A ∪ B) = x(A) ⊕ x(B), (iii) if {Ai}i∈N, Ai ⊆ Ai+1, then x(

i Ai) = i x(Ai).

The least closed subset σ(x) ⊆ R such that x(σ(x)) = 1 is called a spectrum of x.

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Observables

Definition Let E be a monotone σ-complete effect algebra. An observable is a map x : B(R) → E such that (i) x(R) = 1, (ii) if A ∩ B = ∅ then x(A ∪ B) = x(A) ⊕ x(B), (iii) if {Ai}i∈N, Ai ⊆ Ai+1, then x(

i Ai) = i x(Ai).

The least closed subset σ(x) ⊆ R such that x(σ(x)) = 1 is called a spectrum of x. An observable x is bounded if σ(x) ⊆ [a, b], a, b ∈ R,

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Observables

Definition Let E be a monotone σ-complete effect algebra. An observable is a map x : B(R) → E such that (i) x(R) = 1, (ii) if A ∩ B = ∅ then x(A ∪ B) = x(A) ⊕ x(B), (iii) if {Ai}i∈N, Ai ⊆ Ai+1, then x(

i Ai) = i x(Ai).

The least closed subset σ(x) ⊆ R such that x(σ(x)) = 1 is called a spectrum of x. An observable x is bounded if σ(x) ⊆ [a, b], a, b ∈ R, simple if σ(x) is finite.

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Observables

Definition Let E be a monotone σ-complete effect algebra. An observable is a map x : B(R) → E such that (i) x(R) = 1, (ii) if A ∩ B = ∅ then x(A ∪ B) = x(A) ⊕ x(B), (iii) if {Ai}i∈N, Ai ⊆ Ai+1, then x(

i Ai) = i x(Ai).

The least closed subset σ(x) ⊆ R such that x(σ(x)) = 1 is called a spectrum of x. An observable x is bounded if σ(x) ⊆ [a, b], a, b ∈ R, simple if σ(x) is finite. By BO(E), we denote the set of all bounded observables on E.

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Observables

Definition Let E be a monotone σ-complete effect algebra. An observable is a map x : B(R) → E such that (i) x(R) = 1, (ii) if A ∩ B = ∅ then x(A ∪ B) = x(A) ⊕ x(B), (iii) if {Ai}i∈N, Ai ⊆ Ai+1, then x(

i Ai) = i x(Ai).

The least closed subset σ(x) ⊆ R such that x(σ(x)) = 1 is called a spectrum of x. An observable x is bounded if σ(x) ⊆ [a, b], a, b ∈ R, simple if σ(x) is finite. By BO(E), we denote the set of all bounded observables on E.

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Examples

Example Measurable functions (random variables) f : Ω → R on a measure space (Ω, A, p) induce σ-homomorphisms x : B(R) → A by x(B) = f −1(B).

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Examples

Example Measurable functions (random variables) f : Ω → R on a measure space (Ω, A, p) induce σ-homomorphisms x : B(R) → A by x(B) = f −1(B). Example Observables on the prototype effect algebra E(H) bounded positive self-adjoint linear operators on a complex Hilbert space H between 0 and I are (normalized) positive-operator valued measures (POVM).

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)).

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)). Then (2) if t < s, then Bx(t) ≤ Bx(s),

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)). Then (2) if t < s, then Bx(t) ≤ Bx(s), (3)

t<s Bx(t) = Bx(s),

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)). Then (2) if t < s, then Bx(t) ≤ Bx(s), (3)

t<s Bx(t) = Bx(s),

(4)

t∈R Bx(t) = 0, t∈R Bx(t) = 1.

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)). Then (2) if t < s, then Bx(t) ≤ Bx(s), (3)

t<s Bx(t) = Bx(s),

(4)

t∈R Bx(t) = 0, t∈R Bx(t) = 1.

Moreover, for any system {B(t)}t∈R ⊆ E which satisfies (2) – (4) there exists unique observable x on E for which (1) holds.

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Spectral resolutions

Theorem (Dvureˇ censkij, Kukov´ a, 2014) Let x be an observable on a σ-lattice effect algebra E. Set (1) Bx(t) = x((−∞, t)). Then (2) if t < s, then Bx(t) ≤ Bx(s), (3)

t<s Bx(t) = Bx(s),

(4)

t∈R Bx(t) = 0, t∈R Bx(t) = 1.

Moreover, for any system {B(t)}t∈R ⊆ E which satisfies (2) – (4) there exists unique observable x on E for which (1) holds. We call {Bx(t)}t∈R a spectral resolution of x.

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Olson order

Definition (Dvureˇ censkij, 2016) Let x, y ∈ BO(E) be bounded observables on a monotone σ-complete effect algebra E. A relation ≤ on BO(E) given by x ≤ y iff By(t) ≤E Bx(t) for every t ∈ R is a partial order, so called Olson order.

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The sum of observables

Theorem (Dvureˇ censkij, 2016) Let E be a σ-frame effect algebra and let x, y ∈ BO(E). Let: Bx+y(t) :=

• r∈Q

(Bx(r) ∧ By(t − r)) for all t ∈ R. Then there is a unique bounded observable z on E such that Bz(t) = Bx+y(t) for every t ∈ R. We call z the sum of x, y.

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The sum of observables

Theorem (Dvureˇ censkij, 2016) Let E be a σ-frame effect algebra and let x, y ∈ BO(E). Let: Bx+y(t) :=

• r∈Q

(Bx(r) ∧ By(t − r)) for all t ∈ R. Then there is a unique bounded observable z on E such that Bz(t) = Bx+y(t) for every t ∈ R. We call z the sum of x, y. Theorem (Dvureˇ censkij, 2016) Let E be a σ-frame effect algebra and BO(E) the set of all bounded

• bservables on E. Then BO(E) is a distributive lattice and a lattice
• rdered commutative semigroup w.r.t. Olson order and the sum of
• bservables.

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The sum of observables

Theorem (Dvureˇ censkij, 2016) Let E be a σ-frame effect algebra and let x, y ∈ BO(E). Let: Bx+y(t) :=

• r∈Q

(Bx(r) ∧ By(t − r)) for all t ∈ R. Then there is a unique bounded observable z on E such that Bz(t) = Bx+y(t) for every t ∈ R. We call z the sum of x, y. Theorem (Dvureˇ censkij, 2016) Let E be a σ-frame effect algebra and BO(E) the set of all bounded

• bservables on E. Then BO(E) is a distributive lattice and a lattice
• rdered commutative semigroup w.r.t. Olson order and the sum of
• bservables.

In the rest of talk, E will denote a σ-frame effect algebra.

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The sum of observables

Question 1: To find an expression of the sum which is more closely tied to spectra σ(x) and σ(y). Particularly, it should be finite for the case when x, y ∈ BO(E) are simple.

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The sum of observables

Question 1: To find an expression of the sum which is more closely tied to spectra σ(x) and σ(y). Particularly, it should be finite for the case when x, y ∈ BO(E) are simple. Proposition Let x, y ∈ BO(E), σ(x) ⊆ [ax, bx), σ(y) ⊆ [ay, by) for some ax, bx, ay, by ∈ R and let K := Q ∩ (ax ∨ t − cy, t − ay ∧ cx). Then Bx+y(t) =

• r∈K

(Bx(r) ∧ By(t − r)).

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The sum of observables

Question 1: To find an expression of the sum which is more closely tied to spectra σ(x) and σ(y). Particularly, it should be finite for the case when x, y ∈ BO(E) are simple. Proposition Let x, y ∈ BO(E), σ(x) ⊆ [ax, bx), σ(y) ⊆ [ay, by) for some ax, bx, ay, by ∈ R and let K := Q ∩ (ax ∨ t − cy, t − ay ∧ cx). Then Bx+y(t) =

• r∈K

(Bx(r) ∧ By(t − r)). We have Bx(t) = x((∞, t)).

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The sum of observables

Question 1: To find an expression of the sum which is more closely tied to spectra σ(x) and σ(y). Particularly, it should be finite for the case when x, y ∈ BO(E) are simple. Proposition Let x, y ∈ BO(E), σ(x) ⊆ [ax, bx), σ(y) ⊆ [ay, by) for some ax, bx, ay, by ∈ R and let K := Q ∩ (ax ∨ t − cy, t − ay ∧ cx). Then Bx+y(t) =

• r∈K

(Bx(r) ∧ By(t − r)). We have Bx(t) = x((∞, t)). Let us set Bx(t] = x((∞, t]).

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The sum of observables

Theorem Let x, y ∈ B(R). Let M ⊆ R be a subset of R such that for every p ∈ σ(x) there exists Mp ⊆ M such that p = Mp. Then Bx+y(t) =

• m∈M

(Bx(m] ∧ By(t − m)).

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The sum of observables

Theorem Let x, y ∈ B(R). Let M ⊆ R be a subset of R such that for every p ∈ σ(x) there exists Mp ⊆ M such that p = Mp. Then Bx+y(t) =

• m∈M

(Bx(m] ∧ By(t − m)). We can take M = σ(x), i.e., the expression becomes finite when x is a simple observable.

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The sum of observables

Theorem Let x, y ∈ B(R). Let M ⊆ R be a subset of R such that for every p ∈ σ(x) there exists Mp ⊆ M such that p = Mp. Then Bx+y(t) =

• m∈M

(Bx(m] ∧ By(t − m)). We can take M = σ(x), i.e., the expression becomes finite when x is a simple observable. It is worth noting that Bx+y(t) =

• r∈Q

(Bx(r) ∧ By(t − r)) =

• r∈Q

(Bx(r] ∧ By(t − r)) =

• m∈M

(Bx(m] ∧ By(t − m)), but this expression need not to be equal to

m∈M(Bx(m) ∧ By(t − m)).

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The sum of observables

Corollary Let x, y be bounded observables on a σ-distributive lattice effect algebra

• E. Then there exists at most countable set M ⊆ σ(x) such that

Bx+y(t) =

• m∈M

(Bx(m] ∧ By(t − m)).

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The sum of observables

Corollary Let x, y be bounded observables on a σ-distributive lattice effect algebra

• E. Then there exists at most countable set M ⊆ σ(x) such that

Bx+y(t) =

• m∈M

(Bx(m] ∧ By(t − m)). Theorem Let x, y ∈ BO(E). Then Bx+y(t] =

• r∈Q

(Bx(r] ∨ By(t − r)) =

• r∈Q

(Bx(r] ∨ By(t − r]).

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Continuity of the spectral resolution

For an observable x, we have

s<t Bx(s) = Bx(t) for any t ∈ R.

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Continuity of the spectral resolution

For an observable x, we have

s<t Bx(s) = Bx(t) for any t ∈ R.

We say that x has a continuous spectral resolution if for any t ∈ R,

s>t Bx(s) = Bx(t).

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Continuity of the spectral resolution

For an observable x, we have

s<t Bx(s) = Bx(t) for any t ∈ R.

We say that x has a continuous spectral resolution if for any t ∈ R,

s>t Bx(s) = Bx(t).

x has a continuous spectral resolution iff for every t ∈ σ(x), x({t}) = 0.

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Continuity of the spectral resolution

For an observable x, we have

s<t Bx(s) = Bx(t) for any t ∈ R.

We say that x has a continuous spectral resolution if for any t ∈ R,

s>t Bx(s) = Bx(t).

x has a continuous spectral resolution iff for every t ∈ σ(x), x({t}) = 0. Question 2: Preserving of the continuity of spectral resolutions by the sum.

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Continuity of the spectral resolution

For an observable x, we have

s<t Bx(s) = Bx(t) for any t ∈ R.

We say that x has a continuous spectral resolution if for any t ∈ R,

s>t Bx(s) = Bx(t).

x has a continuous spectral resolution iff for every t ∈ σ(x), x({t}) = 0. Question 2: Preserving of the continuity of spectral resolutions by the sum. Theorem Let x, y ∈ BO(E) such as x has a continuous spectral resolution and y is

• simple. Then x + y has a continuous spectral resolution.

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Continuity of the spectral resolution

Let I be a set and {Ji | i ∈ I} a family of non-empty sets. We say that a complete lattice L is completely distributive if for any subset {xij}i∈I,j∈J of L we have

i∈I( j∈J xij) = f :I→J( i∈I xif (j)).

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Continuity of the spectral resolution

Let I be a set and {Ji | i ∈ I} a family of non-empty sets. We say that a complete lattice L is completely distributive if for any subset {xij}i∈I,j∈J of L we have

i∈I( j∈J xij) = f :I→J( i∈I xif (j)).

Theorem Let x, y ∈ BO(E) be bounded observables on a completely distributive lattice effect algebra E. If x has a continuous spectral resolution, then x + y has a continuous spectral resolution.

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Extremal points of the spectrum of the sum

Question 3: To show how is the spectrum σ(x + y) connected with spectra σ(x) and σ(y).

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Extremal points of the spectrum of the sum

Question 3: To show how is the spectrum σ(x + y) connected with spectra σ(x) and σ(y). Theorem Let x, y ∈ BO(E). Then σ(x + y) ⊆ σ(x) + σ(y).

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Extremal points of the spectrum of the sum

Question 3: To show how is the spectrum σ(x + y) connected with spectra σ(x) and σ(y). Theorem Let x, y ∈ BO(E). Then σ(x + y) ⊆ σ(x) + σ(y). Corollary Let x, y ∈ BO(E) such that |σ(x)| = m and |σ(y)| = n. Then σ(x + y) ⊆ {r + s | r ∈ σ(x), s ∈ σ(y)}, that is, |σ(x + y)| ≤ m · n. An example of x, y ∈ BO(E) such that |σ(x + y)| = m · n exists.

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Extremal points of the spectrum of the sum

Theorem Let x, y ∈ BO(E). TFAE: 1.) if p, q ∈ Q and Bx(p) > 0, By(q) > 0, then Bx(p) ∧ By(q) > 0, 2.) σ(x) + σ(y) ∈ σ(x + y), 3.) σ(x) + σ(y) = σ(x + y). Moreover, TFAE: 1.) if p, q ∈ Q, p < σ(x), q < σ(y), then Bx(p) ∨ By(q) < 1, 2.) σ(x) + σ(y) ∈ σ(x + y), 3.) σ(x) + σ(y) = σ(x + y).

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Inverse elements

Let f : R → R be a mapping defined by f (t) := −t. Define −x : B(R) → E by −x(A) := x(f (A)),

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Inverse elements

Let f : R → R be a mapping defined by f (t) := −t. Define −x : B(R) → E by −x(A) := x(f (A)), If x(A) ∧ x(A)′ = 0 for every A ∈ B(R), then we call x a sharp

• bservable.

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Inverse elements

Let f : R → R be a mapping defined by f (t) := −t. Define −x : B(R) → E by −x(A) := x(f (A)), If x(A) ∧ x(A)′ = 0 for every A ∈ B(R), then we call x a sharp

• bservable.

Theorem (Dvureˇ censkij 2016) Let E be a σ-frame effect algebra. The set of sharp bounded observables SBO(E) ⊆ BO(E) is with respect to Olson order and the sum of

• bservables a lattice-ordered group in which −x is the inverse element of x

and the neutral element q0 is given by σ(q0) = {0}. Moreover, SBO(E) is a subsemigroup and a sublattice of the lattice-ordered semigroup BO(E).

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Inverse elements

Let f : R → R be a mapping defined by f (t) := −t. Define −x : B(R) → E by −x(A) := x(f (A)), If x(A) ∧ x(A)′ = 0 for every A ∈ B(R), then we call x a sharp

• bservable.

Theorem (Dvureˇ censkij 2016) Let E be a σ-frame effect algebra. The set of sharp bounded observables SBO(E) ⊆ BO(E) is with respect to Olson order and the sum of

• bservables a lattice-ordered group in which −x is the inverse element of x

and the neutral element q0 is given by σ(q0) = {0}. Moreover, SBO(E) is a subsemigroup and a sublattice of the lattice-ordered semigroup BO(E). Theorem Let x, y ∈ BO(E). Then (−x) + (−y) = −(x + y).

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Decomposition to a sharp and a meager part

By ˆ x (˜ x respectively) we denote the least (greatest) sharp observable greater (less) than x. We say that x is a meager observable if σ(˜ x) = {a} and a dense observable if σ(ˆ x) = {b}.

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Decomposition to a sharp and a meager part

By ˆ x (˜ x respectively) we denote the least (greatest) sharp observable greater (less) than x. We say that x is a meager observable if σ(˜ x) = {a} and a dense observable if σ(ˆ x) = {b}. Theorem Let x ∈ BO(E). Then x = ˜ x + xm = ˆ x + xd where xm is a meager

• bservable and xd a dense observable. Moreover, we have xm = −(−x)d

and σ(xm) = σ(xd) = 0.

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Decomposition to a sharp and a meager part

By ˆ x (˜ x respectively) we denote the least (greatest) sharp observable greater (less) than x. We say that x is a meager observable if σ(˜ x) = {a} and a dense observable if σ(ˆ x) = {b}. Theorem Let x ∈ BO(E). Then x = ˜ x + xm = ˆ x + xd where xm is a meager

• bservable and xd a dense observable. Moreover, we have xm = −(−x)d

and σ(xm) = σ(xd) = 0. Theorem Subsets M0BO(E), D0BO(E) ⊆ BO(E) of meager, resp. dense,

• bservables such that σ(xm) = 0, resp. σ(xd) = 0, forms

subsemigroups and sublattices of the lattice-ordered semigroup BO(E). Moreover, BO(E) ∼ = SBO(E) ⊕ M0BO(E) ∼ = D0BO(E) ⊕ M0BO(E).

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References

Dvureˇ censkij, A., Pulmannov´ a, S.: “New Trends in Quantum Structures”, Kluwer Acad. Publ., Dotrecht, Ister Science, Bratislava, 2000, 541 + xvi pp. Dvureˇ censkij, A., Kukov´ a, M: Observables on Quantum Structures,

• Inf. Sci. 262, 215–222 (2014).

Dvureˇ censkij, A.:Olson Order of Quantum Observables, Int. J. Theor.

• Phys. 55, 4896–4912 (2016).

Dvureˇ censkij, A.: Sum of observables on MV-effect algebras, Soft Computing DOI: 10.1007/s00500-017-2741-1. Janda, J., Xie, Y.: The spectrum of the sum of observables on σ-complete MV-effect algebras, Soft Comput. (2018), https://doi.org/10.1007/s00500-018-3078-0.

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Thank you for your attention!

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