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Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions...
Transcript of Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions...
![Page 1: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/1.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra of commutative BCK-algebras
Matt Evans
Binghamton [email protected]
BLAST 2019University of Colorado, Boulder
May 23, 2019
![Page 2: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/2.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Overview
1 Definition, facts, examples
2 Ideals
3 Spectra
4 Connected Unions
5 Current/Future Work
![Page 3: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/3.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
cBCK-algebras
A commutative BCK-algebra A is an algebra A = 〈A; ·, 0〉 oftype (2, 0) such that
(x · y) · z = (x · z) · yx · (x · y) = y · (y · x)
x · x = 0
x · 0 = x
for all x , y , z ∈ A.
![Page 4: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/4.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
cBCK-algebras
A commutative BCK-algebra A is an algebra A = 〈A; ·, 0〉 oftype (2, 0) such that
(x · y) · z = (x · z) · yx · (x · y) = y · (y · x)
x · x = 0
x · 0 = x
for all x , y , z ∈ A.
![Page 5: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/5.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
cBCK-algebras
A commutative BCK-algebra A is an algebra A = 〈A; ·, 0〉 oftype (2, 0) such that
(x · y) · z = (x · z) · yx · (x · y) = y · (y · x)
x · x = 0
x · 0 = x
for all x , y , z ∈ A.
![Page 6: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/6.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Facts
1 Every BCK-algebra is partially ordered via: x ≤ y iff x · y = 0.
2 Commutativity: x ∧ y := y · (y · x) = x · (x · y) = y ∧ x .
A BCK-algebra A is bounded if there exists an element 1 ∈ Asuch that x · 1 = 0 for all x ∈ A.
3 If A is bounded, then 〈A;∧,∨〉 is a distributive lattice, where
x ∨ y := 1 ·[(1 · x) ∧ (1 · y)
].
(Traczyk, 1979)
![Page 7: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/7.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Facts
1 Every BCK-algebra is partially ordered via: x ≤ y iff x · y = 0.
2 Commutativity: x ∧ y := y · (y · x)
= x · (x · y) = y ∧ x .
A BCK-algebra A is bounded if there exists an element 1 ∈ Asuch that x · 1 = 0 for all x ∈ A.
3 If A is bounded, then 〈A;∧,∨〉 is a distributive lattice, where
x ∨ y := 1 ·[(1 · x) ∧ (1 · y)
].
(Traczyk, 1979)
![Page 8: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/8.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Facts
1 Every BCK-algebra is partially ordered via: x ≤ y iff x · y = 0.
2 Commutativity: x ∧ y := y · (y · x) = x · (x · y) = y ∧ x .
A BCK-algebra A is bounded if there exists an element 1 ∈ Asuch that x · 1 = 0 for all x ∈ A.
3 If A is bounded, then 〈A;∧,∨〉 is a distributive lattice, where
x ∨ y := 1 ·[(1 · x) ∧ (1 · y)
].
(Traczyk, 1979)
![Page 9: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/9.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Facts
1 Every BCK-algebra is partially ordered via: x ≤ y iff x · y = 0.
2 Commutativity: x ∧ y := y · (y · x) = x · (x · y) = y ∧ x .
A BCK-algebra A is bounded if there exists an element 1 ∈ Asuch that x · 1 = 0 for all x ∈ A.
3 If A is bounded, then 〈A;∧,∨〉 is a distributive lattice, where
x ∨ y := 1 ·[(1 · x) ∧ (1 · y)
].
(Traczyk, 1979)
![Page 10: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/10.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Facts
1 Every BCK-algebra is partially ordered via: x ≤ y iff x · y = 0.
2 Commutativity: x ∧ y := y · (y · x) = x · (x · y) = y ∧ x .
A BCK-algebra A is bounded if there exists an element 1 ∈ Asuch that x · 1 = 0 for all x ∈ A.
3 If A is bounded, then 〈A;∧,∨〉 is a distributive lattice, where
x ∨ y := 1 ·[(1 · x) ∧ (1 · y)
].
(Traczyk, 1979)
![Page 11: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/11.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Let X be a set. Then⟨P(X );−,∅
⟩is a cBCK-algebra.
2 More generally, any Boolean algebra B admits acBCK-structure via x · y = x ∧ (¬y).
3 R≥0 = 〈R≥0; ·, 0〉 is a cBCK-algebra, wherex · y = max{x − y , 0} .
4 N0 is a subalgebra of R≥0
5 Let O = N0⊕N∂0 denote the ordinal sum of N0 with itsorder-dual. Then O admits a cBCK-structure.
![Page 12: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/12.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Let X be a set. Then⟨P(X );−,∅
⟩is a cBCK-algebra.
2 More generally, any Boolean algebra B admits acBCK-structure via x · y = x ∧ (¬y).
3 R≥0 = 〈R≥0; ·, 0〉 is a cBCK-algebra, wherex · y = max{x − y , 0} .
4 N0 is a subalgebra of R≥0
5 Let O = N0⊕N∂0 denote the ordinal sum of N0 with itsorder-dual. Then O admits a cBCK-structure.
![Page 13: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/13.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Let X be a set. Then⟨P(X );−,∅
⟩is a cBCK-algebra.
2 More generally, any Boolean algebra B admits acBCK-structure via x · y = x ∧ (¬y).
3 R≥0 = 〈R≥0; ·, 0〉 is a cBCK-algebra, wherex · y = max{x − y , 0} .
4 N0 is a subalgebra of R≥0
5 Let O = N0⊕N∂0 denote the ordinal sum of N0 with itsorder-dual. Then O admits a cBCK-structure.
![Page 14: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/14.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Let X be a set. Then⟨P(X );−,∅
⟩is a cBCK-algebra.
2 More generally, any Boolean algebra B admits acBCK-structure via x · y = x ∧ (¬y).
3 R≥0 = 〈R≥0; ·, 0〉 is a cBCK-algebra, wherex · y = max{x − y , 0} .
4 N0 is a subalgebra of R≥0
5 Let O = N0⊕N∂0 denote the ordinal sum of N0 with itsorder-dual. Then O admits a cBCK-structure.
![Page 15: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/15.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Let X be a set. Then⟨P(X );−,∅
⟩is a cBCK-algebra.
2 More generally, any Boolean algebra B admits acBCK-structure via x · y = x ∧ (¬y).
3 R≥0 = 〈R≥0; ·, 0〉 is a cBCK-algebra, wherex · y = max{x − y , 0} .
4 N0 is a subalgebra of R≥0
5 Let O = N0⊕N∂0 denote the ordinal sum of N0 with itsorder-dual. Then O admits a cBCK-structure.
![Page 16: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/16.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example
Let {Aλ}λ∈Λ be a family of cBCK-algebras with Aλ ∩ Aµ = {0}.
Put U =⋃λ∈Λ
Aλ.
Then U is a cBCK-algebra with the operation
x · y =
{x ·λ y if x , y ∈ Aλ
x otherwise .
We call U the BCK-union of the Aλ’s and we will write U =⊔Aλ
![Page 17: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/17.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example
Let {Aλ}λ∈Λ be a family of cBCK-algebras with Aλ ∩ Aµ = {0}.
Put U =⋃λ∈Λ
Aλ.
Then U is a cBCK-algebra with the operation
x · y =
{x ·λ y if x , y ∈ Aλ
x otherwise .
We call U the BCK-union of the Aλ’s and we will write U =⊔Aλ
![Page 18: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/18.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example
Let {Aλ}λ∈Λ be a family of cBCK-algebras with Aλ ∩ Aµ = {0}.
Put U =⋃λ∈Λ
Aλ.
Then U is a cBCK-algebra with the operation
x · y =
{x ·λ y if x , y ∈ Aλ
x otherwise .
We call U the BCK-union of the Aλ’s and we will write U =⊔Aλ
![Page 19: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/19.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example
Let {Aλ}λ∈Λ be a family of cBCK-algebras with Aλ ∩ Aµ = {0}.
Put U =⋃λ∈Λ
Aλ.
Then U is a cBCK-algebra with the operation
x · y =
{x ·λ y if x , y ∈ Aλ
x otherwise .
We call U the BCK-union of the Aλ’s and we will write U =⊔Aλ
![Page 20: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/20.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Ideals
Given a cBCK-algebra A, a subset I is an ideal if
1 0 ∈ I
2 if x · y ∈ I and y ∈ I , then x ∈ I .
A proper ideal P of A is a prime ideal if x ∧ y ∈ P implies x ∈ Por y ∈ P.
Let Id(A) and X(A) denote the set of ideals and prime ideals,respectively.
![Page 21: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/21.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Ideals
Given a cBCK-algebra A, a subset I is an ideal if
1 0 ∈ I
2 if x · y ∈ I and y ∈ I , then x ∈ I .
A proper ideal P of A is a prime ideal if x ∧ y ∈ P implies x ∈ Por y ∈ P.
Let Id(A) and X(A) denote the set of ideals and prime ideals,respectively.
![Page 22: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/22.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Ideals
Given a cBCK-algebra A, a subset I is an ideal if
1 0 ∈ I
2 if x · y ∈ I and y ∈ I , then x ∈ I .
A proper ideal P of A is a prime ideal if x ∧ y ∈ P implies x ∈ Por y ∈ P.
Let Id(A) and X(A) denote the set of ideals and prime ideals,respectively.
![Page 23: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/23.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Id(N0) ={{0},N0
}and X(N0) =
{{0}}
.
2 Id(O) ={{0},N0,O
}and X(O) =
{{0},N0,
}.
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Theorem (E., 2019)
An ideal P in a BCK-union U is prime iff there exists µ ∈ Λ andQ ∈ X(Aµ) such that
P =⊔
AQλ,µ ,
where AQλ,µ =
{Aλ if λ 6= µ
Q if λ = µ .
![Page 24: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/24.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Id(N0) ={{0},N0
}and X(N0) =
{{0}}
.
2 Id(O) ={{0},N0,O
}and X(O) =
{{0},N0,
}.
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Theorem (E., 2019)
An ideal P in a BCK-union U is prime iff there exists µ ∈ Λ andQ ∈ X(Aµ) such that
P =⊔
AQλ,µ ,
where AQλ,µ =
{Aλ if λ 6= µ
Q if λ = µ .
![Page 25: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/25.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Id(N0) ={{0},N0
}and X(N0) =
{{0}}
.
2 Id(O) ={{0},N0,O
}and X(O) =
{{0},N0,
}.
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Theorem (E., 2019)
An ideal P in a BCK-union U is prime iff there exists µ ∈ Λ andQ ∈ X(Aµ) such that
P =⊔
AQλ,µ ,
where AQλ,µ =
{Aλ if λ 6= µ
Q if λ = µ .
![Page 26: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/26.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 Id(N0) ={{0},N0
}and X(N0) =
{{0}}
.
2 Id(O) ={{0},N0,O
}and X(O) =
{{0},N0,
}.
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Theorem (E., 2019)
An ideal P in a BCK-union U is prime iff there exists µ ∈ Λ andQ ∈ X(Aµ) such that
P =⊔
AQλ,µ ,
where AQλ,µ =
{Aλ if λ 6= µ
Q if λ = µ .
![Page 27: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/27.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra
For a subset S ⊆ A, define
σ(S) ={P ∈ X(A) | S 6⊆ P
}.
We will write σ(a) for σ({a}).
Proposition
The family T(A) ={σ(I ) | I ∈ Id(A)
}is a topology on X(A), and
T0(A) ={σ(a) | a ∈ A
}is a basis for this topology.
The space(X(A) , T(A)
)is the spectrum of A.
![Page 28: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/28.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra
For a subset S ⊆ A, define
σ(S) ={P ∈ X(A) | S 6⊆ P
}.
We will write σ(a) for σ({a}).
Proposition
The family T(A) ={σ(I ) | I ∈ Id(A)
}is a topology on X(A), and
T0(A) ={σ(a) | a ∈ A
}is a basis for this topology.
The space(X(A) , T(A)
)is the spectrum of A.
![Page 29: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/29.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra
For a subset S ⊆ A, define
σ(S) ={P ∈ X(A) | S 6⊆ P
}.
We will write σ(a) for σ({a}).
Proposition
The family T(A) ={σ(I ) | I ∈ Id(A)
}is a topology on X(A),
andT0(A) =
{σ(a) | a ∈ A
}is a basis for this topology.
The space(X(A) , T(A)
)is the spectrum of A.
![Page 30: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/30.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra
For a subset S ⊆ A, define
σ(S) ={P ∈ X(A) | S 6⊆ P
}.
We will write σ(a) for σ({a}).
Proposition
The family T(A) ={σ(I ) | I ∈ Id(A)
}is a topology on X(A), and
T0(A) ={σ(a) | a ∈ A
}is a basis for this topology.
The space(X(A) , T(A)
)is the spectrum of A.
![Page 31: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/31.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Spectra
For a subset S ⊆ A, define
σ(S) ={P ∈ X(A) | S 6⊆ P
}.
We will write σ(a) for σ({a}).
Proposition
The family T(A) ={σ(I ) | I ∈ Id(A)
}is a topology on X(A), and
T0(A) ={σ(a) | a ∈ A
}is a basis for this topology.
The space(X(A) , T(A)
)is the spectrum of A.
![Page 32: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/32.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
A topological space (X ,T) is a Stone space if it is compact,Hausdorff, and totally disconnected.
An ordered topological space (X ,T,≤) is called a Priestley spaceif (X ,T) is a Stone space and (X ,≤) satisfies the followingseparation property:
if x 6≤ y , there exists a clopen up-set U such that x ∈ U but y /∈ U .
Theorem (Meng & Jun, 1998)
If A is a bounded cBCK-algebra, then X(A) is a Stone space.
![Page 33: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/33.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
A topological space (X ,T) is a Stone space if it is compact,Hausdorff, and totally disconnected.
An ordered topological space (X ,T,≤) is called a Priestley spaceif (X ,T) is a Stone space and (X ,≤) satisfies the followingseparation property:
if x 6≤ y , there exists a clopen up-set U such that x ∈ U but y /∈ U .
Theorem (Meng & Jun, 1998)
If A is a bounded cBCK-algebra, then X(A) is a Stone space.
![Page 34: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/34.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
A topological space (X ,T) is a Stone space if it is compact,Hausdorff, and totally disconnected.
An ordered topological space (X ,T,≤) is called a Priestley spaceif (X ,T) is a Stone space and (X ,≤) satisfies the followingseparation property:
if x 6≤ y , there exists a clopen up-set U such that x ∈ U but y /∈ U .
Theorem (Meng & Jun, 1998)
If A is a bounded cBCK-algebra, then X(A) is a Stone space.
![Page 35: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/35.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Theorem (E., 2017)
If A is a bounded and involutory cBCK-algebra, then X(A) is aPriestley space.
A BCK-algebra A is involutory if, for every pair of elementsx , y ∈ A, the decreasing sequence
x ≥ x · y ≥ (x · y) · y ≥((x · y) · y
)· y ≥ · · ·
eventually stabilizes.
Proposition (E., 2018)
X(A) is locally compact for any cBCK-algebra A.
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
![Page 36: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/36.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Theorem (E., 2017)
If A is a bounded and involutory cBCK-algebra, then X(A) is aPriestley space.
A BCK-algebra A is involutory if, for every pair of elementsx , y ∈ A, the decreasing sequence
x ≥ x · y ≥ (x · y) · y ≥((x · y) · y
)· y ≥ · · ·
eventually stabilizes.
Proposition (E., 2018)
X(A) is locally compact for any cBCK-algebra A.
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
![Page 37: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/37.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Theorem (E., 2017)
If A is a bounded and involutory cBCK-algebra, then X(A) is aPriestley space.
A BCK-algebra A is involutory if, for every pair of elementsx , y ∈ A, the decreasing sequence
x ≥ x · y ≥ (x · y) · y ≥((x · y) · y
)· y ≥ · · ·
eventually stabilizes.
Proposition (E., 2018)
X(A) is locally compact for any cBCK-algebra A.
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
![Page 38: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/38.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Theorem (E., 2017)
If A is a bounded and involutory cBCK-algebra, then X(A) is aPriestley space.
A BCK-algebra A is involutory if, for every pair of elementsx , y ∈ A, the decreasing sequence
x ≥ x · y ≥ (x · y) · y ≥((x · y) · y
)· y ≥ · · ·
eventually stabilizes.
Proposition (E., 2018)
X(A) is locally compact for any cBCK-algebra A.
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
![Page 39: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/39.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
Open Question
Is A being involutory a necessary condition for X(A) to satisfyPriestley separation?
Open Question
What is the precise relationship between boundedness of A andcompactness of X(A)? Specifically, is there an unbounded A withinfinite compact spectrum?
Open Question
What is the image of the functor X : cBCK→ Top ?
![Page 40: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/40.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
Open Question
Is A being involutory a necessary condition for X(A) to satisfyPriestley separation?
Open Question
What is the precise relationship between boundedness of A andcompactness of X(A)? Specifically, is there an unbounded A withinfinite compact spectrum?
Open Question
What is the image of the functor X : cBCK→ Top ?
![Page 41: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/41.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
Open Question
Is A being involutory a necessary condition for X(A) to satisfyPriestley separation?
Open Question
What is the precise relationship between boundedness of A andcompactness of X(A)?
Specifically, is there an unbounded A withinfinite compact spectrum?
Open Question
What is the image of the functor X : cBCK→ Top ?
![Page 42: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/42.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
Open Question
Is A being involutory a necessary condition for X(A) to satisfyPriestley separation?
Open Question
What is the precise relationship between boundedness of A andcompactness of X(A)? Specifically, is there an unbounded A withinfinite compact spectrum?
Open Question
What is the image of the functor X : cBCK→ Top ?
![Page 43: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/43.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Comm Comm, Bdd Comm, Invol Comm, Bdd, Invol
locally compact compact locally compact compactHausdorff Hausdorff Hausdorff
t.d. t.d. t.d.Pries. sep. Pries. sep.
Open Question
Is A being involutory a necessary condition for X(A) to satisfyPriestley separation?
Open Question
What is the precise relationship between boundedness of A andcompactness of X(A)? Specifically, is there an unbounded A withinfinite compact spectrum?
Open Question
What is the image of the functor X : cBCK→ Top ?
![Page 44: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/44.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 X(N0
)is a one-point space.
2 X(O)
is the Sierpinski space. [Reminder: O = N0⊕N∂0 . ]
3 X( ⊔λ∈Λ
N0
)is a discrete space of cardinality |Λ|.
4 X( ⊔λ∈Λ
O)
is actually interesting!
![Page 45: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/45.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 X(N0
)is a one-point space.
2 X(O)
is the Sierpinski space. [Reminder: O = N0⊕N∂0 . ]
3 X( ⊔λ∈Λ
N0
)is a discrete space of cardinality |Λ|.
4 X( ⊔λ∈Λ
O)
is actually interesting!
![Page 46: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/46.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 X(N0
)is a one-point space.
2 X(O)
is the Sierpinski space. [Reminder: O = N0⊕N∂0 . ]
3 X( ⊔λ∈Λ
N0
)is a discrete space of cardinality |Λ|.
4 X( ⊔λ∈Λ
O)
is actually interesting!
![Page 47: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/47.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples
1 X(N0
)is a one-point space.
2 X(O)
is the Sierpinski space. [Reminder: O = N0⊕N∂0 . ]
3 X( ⊔λ∈Λ
N0
)is a discrete space of cardinality |Λ|.
4 X( ⊔λ∈Λ
O)
is actually interesting!
![Page 48: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/48.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Consider A = O1⊔
O2 and for brevity let X = X(A). The primeideals of A are:
Z1 = {0} t O2 N1 = N0 t O2
Z2 = O1 t {0} N2 = O1 t N0
So X is a 4-point space and the topology is:
![Page 49: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/49.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Consider A = O1⊔
O2 and for brevity let X = X(A). The primeideals of A are:
Z1 = {0} t O2 N1 = N0 t O2
Z2 = O1 t {0} N2 = O1 t N0
So X is a 4-point space and the topology is:
![Page 50: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/50.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space ; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 51: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/51.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space ; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 52: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/52.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space ; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 53: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/53.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space ; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 54: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/54.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space
; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 55: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/55.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Example: O1
⊔O2
Some facts:
1 X is not Hausdorff.
2 Every irreducible closed set is the closure of a unique point;that is, X is a sober space.
3 The compact open sets form a basis and are closed underfinite intersections.
4 X is compact.
Properties (2)-(4) tell us that X is a spectral space ; that is, X ishomeomorphic to the spectrum of some commutative ring.
![Page 56: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/56.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples:n⊔
i=1
Oi and⊔λ∈Λ
Oλ
More generally, the spectrum ofn⊔
i=1Oi is a non-Hausdorff spectral
space with cardinality 2n.
If we let Λ be an infinite indexing set, then the spectrum of⊔λ∈Λ
Oλ
is no longer compact, but still has properties (2) and (3). Thismakes it a generalized spectral space, meaning it ishomeomorphic to the spectrum of some distributive lattice withzero.
![Page 57: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/57.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples:n⊔
i=1
Oi and⊔λ∈Λ
Oλ
More generally, the spectrum ofn⊔
i=1Oi is a non-Hausdorff spectral
space with cardinality 2n.
If we let Λ be an infinite indexing set, then the spectrum of⊔λ∈Λ
Oλ
is no longer compact, but still has properties (2) and (3).
Thismakes it a generalized spectral space, meaning it ishomeomorphic to the spectrum of some distributive lattice withzero.
![Page 58: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/58.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples:n⊔
i=1
Oi and⊔λ∈Λ
Oλ
More generally, the spectrum ofn⊔
i=1Oi is a non-Hausdorff spectral
space with cardinality 2n.
If we let Λ be an infinite indexing set, then the spectrum of⊔λ∈Λ
Oλ
is no longer compact, but still has properties (2) and (3). Thismakes it a generalized spectral space, meaning it ishomeomorphic to the spectrum of some distributive lattice withzero.
![Page 59: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/59.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Examples:n⊔
i=1
Oi and⊔λ∈Λ
Oλ
More generally, the spectrum ofn⊔
i=1Oi is a non-Hausdorff spectral
space with cardinality 2n.
If we let Λ be an infinite indexing set, then the spectrum of⊔λ∈Λ
Oλ
is no longer compact, but still has properties (2) and (3). Thismakes it a generalized spectral space, meaning it ishomeomorphic to the spectrum of some distributive lattice withzero.
![Page 60: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/60.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Some associated facts:
Proposition (E., 2019)
X(⊔
Aλ)∼=⊔X(Aλ) with the disjoint union topology.
Proposition (E., 2019)
X( n∏i=1
Ai
)∼= X
( n⊔i=1
Ai
)
![Page 61: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/61.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Some associated facts:
Proposition (E., 2019)
X(⊔
Aλ)∼=⊔
X(Aλ) with the disjoint union topology.
Proposition (E., 2019)
X( n∏i=1
Ai
)∼= X
( n⊔i=1
Ai
)
![Page 62: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/62.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Some associated facts:
Proposition (E., 2019)
X(⊔
Aλ)∼=⊔
X(Aλ) with the disjoint union topology.
Proposition (E., 2019)
X( n∏i=1
Ai
)∼= X
( n⊔i=1
Ai
)
![Page 63: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/63.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 64: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/64.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 65: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/65.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 66: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/66.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 67: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/67.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 68: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/68.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
A family {Aλ}λ∈Λ of BCK-algebras is connected if for allλ, µ ∈ Λ:
1 The operations ·λ and ·µ coincide on Aλ ∩ Aµ, and
2 if x ∈ Aλ \ Aµ and y ∈ Aµ \ Aλ, then glb{x , y} ∈ Aλ ∩ Aµ.
Given a connected family A = {Aλ}λ∈Λ, we say A = 〈A; ·, 0〉 is aconnected BCK-union of the family A provided
1 A =⋃λ∈Λ
Aλ, and
2 the operation · agrees with ·λ when restricted to Aλ.
Note: the BCK-union we defined earlier is a special case ofconnected BCK-union with Aλ ∩ Aµ = {0} for all pairs of indices.
Note: Any family of commutative BCK-algebras satisfying (1)automatically satisfies (2).
![Page 69: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/69.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
Theorem (Pa lasinski, 1982)
1 The connected BCK-union for every connected family ofBCK-algebras exists and it is unique.
2 If {Aλ}λ∈Λ is a connected family of commutativeBCK-algebras, then their connected BCK-union is acommutative BCK-algebra.
3 Every cBCK-algebra is the connected BCK-union of aconnected family of directed cBCK-algebras.
![Page 70: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/70.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
Theorem (Pa lasinski, 1982)
1 The connected BCK-union for every connected family ofBCK-algebras exists and it is unique.
2 If {Aλ}λ∈Λ is a connected family of commutativeBCK-algebras, then their connected BCK-union is acommutative BCK-algebra.
3 Every cBCK-algebra is the connected BCK-union of aconnected family of directed cBCK-algebras.
![Page 71: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/71.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
Theorem (Pa lasinski, 1982)
1 The connected BCK-union for every connected family ofBCK-algebras exists and it is unique.
2 If {Aλ}λ∈Λ is a connected family of commutativeBCK-algebras, then their connected BCK-union is acommutative BCK-algebra.
3 Every cBCK-algebra is the connected BCK-union of aconnected family of directed cBCK-algebras.
![Page 72: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/72.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Connected Unions
Theorem (Pa lasinski, 1982)
1 The connected BCK-union for every connected family ofBCK-algebras exists and it is unique.
2 If {Aλ}λ∈Λ is a connected family of commutativeBCK-algebras, then their connected BCK-union is acommutative BCK-algebra.
3 Every cBCK-algebra is the connected BCK-union of aconnected family of directed cBCK-algebras.
![Page 73: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/73.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Current/Future Work
Reminder:
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Current Can we decompose the ideals of a connected BCK-union insome “nice” way that is similar to the BCK-union?
Future Same question for prime ideals.
![Page 74: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/74.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Current/Future Work
Reminder:
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Current Can we decompose the ideals of a connected BCK-union insome “nice” way that is similar to the BCK-union?
Future Same question for prime ideals.
![Page 75: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/75.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Current/Future Work
Reminder:
Lemma
If U =⊔Aλ is a BCK-union, then any ideal has the form I =
⊔Iλ,
where Iλ ∈ Id(Aλ).
Current Can we decompose the ideals of a connected BCK-union insome “nice” way that is similar to the BCK-union?
Future Same question for prime ideals.
![Page 76: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/76.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
Thank you!
![Page 77: Spectra of commutative BCK-algebras · De nition, facts, examples Ideals Spectra Connected Unions Current/Future Work cBCK-algebras A commutative BCK-algebra A is an algebra A = hA;](https://reader034.fdocuments.us/reader034/viewer/2022050609/5fb09c494b4e1435f53c6af7/html5/thumbnails/77.jpg)
Definition, facts, examples Ideals Spectra Connected Unions Current/Future Work
References
Iseki, Tanaka
An introduction to the theory of BCK-algebras
Mathematica Japonica, 1978, 23 (1), 1-26.
Meng, Jun
The spectral space of MV-algebras is a Stone space
Scientiae Mathematicae, 1998, 1 (2), 211-215.
Pa lasinski
Representation theorem for commutative BCK-algebras
Mathematics Seminar Notes, Kobe University, 1982, 10, 473-478.
Traczyk
On the variety of bounded commutative BCK-algebras
Mathematica Japonica, 1979, 24 (3), 283-292.