the homomorphism lattice induced by a finite algebra...some examples i the case where c is the...
TRANSCRIPT
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The homomorphism latticeinduced by a finite algebra
Brian A. Davey, Charles T. Gray and Jane G. Pitkethly
AAA94+NSAC 2017
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The homomorphism order on a category
I Let C be a category.I Define a quasi-order→ on the objects of C by X→ Y if
there exists a morphism from X to Y.I Now set
X ≡ Y ⇐⇒ X→ Y and Y→ X.
Then→ induces an order on the class C/≡.
I To ensure that C/≡ is a set we take C to be a class of finitestructures with all homomorphisms between them.
I Then we can define the ordered set
PC := 〈C/≡;→〉,
which we refer to as the homomorphism order on C.
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The homomorphism order on a category
I Let C be a category.I Define a quasi-order→ on the objects of C by X→ Y if
there exists a morphism from X to Y.I Now set
X ≡ Y ⇐⇒ X→ Y and Y→ X.
Then→ induces an order on the class C/≡.I To ensure that C/≡ is a set we take C to be a class of finite
structures with all homomorphisms between them.I Then we can define the ordered set
PC := 〈C/≡;→〉,
which we refer to as the homomorphism order on C.
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Categories with product and coproduct
I PC := 〈C/≡;→〉, where X ≡ Y ⇐⇒ X→ Y and Y→ X.
I If the category C has pairwise products and coproducts,then PC is a lattice:
I the meet of X/≡ and Y/≡ is (X× Y)/≡,I the join of X/≡ and Y/≡ is (X t Y)/≡.
X× Y
X Y
X t Y
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Some examples
I The case where C is the category of finite graphs orfinite digraphs has been studied extensively:
I in both cases, PC = 〈C/≡;→〉 is a distributive lattice(in fact, a Heyting algebra) into which every countableordered set can be embedded.
I We are interested in the case where C is a category offinite algebras.
I For many natural categories of finite algebras, the orderedset PC is trivial as there are homomorphisms betweenevery pair of objects in C. For example:
I groups, vector spaces over a finite field, rings, lattices,semigroups.
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Some examples
I The case where C is the category of finite graphs orfinite digraphs has been studied extensively:
I in both cases, PC = 〈C/≡;→〉 is a distributive lattice(in fact, a Heyting algebra) into which every countableordered set can be embedded.
I We are interested in the case where C is a category offinite algebras.
I For many natural categories of finite algebras, the orderedset PC is trivial as there are homomorphisms betweenevery pair of objects in C. For example:
I groups, vector spaces over a finite field, rings, lattices,semigroups.
![Page 7: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/7.jpg)
Some examples
I The case where C is the category of finite graphs orfinite digraphs has been studied extensively:
I in both cases, PC = 〈C/≡;→〉 is a distributive lattice(in fact, a Heyting algebra) into which every countableordered set can be embedded.
I We are interested in the case where C is a category offinite algebras.
I For many natural categories of finite algebras, the orderedset PC is trivial as there are homomorphisms betweenevery pair of objects in C. For example:
I groups, vector spaces over a finite field, rings, lattices,semigroups.
![Page 8: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/8.jpg)
Locally finite varieties
I Let A be a variety of algebras and let Afin denote theclass of finite members of A.
I If Afin is closed under forming finite coproducts,then PAfin is a lattice.
I Hence, it is natural to look at locally finite varieties.
LemmaLet A be a locally finite variety. Then the homomorphism orderPAfin = 〈Afin/≡;→〉 is a countable bounded lattice.
NoteI The top of PAfin is the equivalence class consisting of
algebras in Afin that have a trivial subalgebra.I The bottom of PAfin is the equivalence class that contains
all finitely generated free algebras in A.
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The homomorphism lattice induced by a finite algebra
FactLet A be a finite algebra. Then the variety Var(A) generated byA is locally finite.
NotationGiven a finite algebra A, we define the lattice
LA := 〈Var(A)fin/≡;→〉
and refer to it as the homomorphism lattice induced by A.
QuestionI Which lattices arise as LA for some finite algebra A?
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The homomorphism lattice induced by a finite algebra
FactLet A be a finite algebra. Then the variety Var(A) generated byA is locally finite.
NotationGiven a finite algebra A, we define the lattice
LA := 〈Var(A)fin/≡;→〉
and refer to it as the homomorphism lattice induced by A.
QuestionI Which lattices arise as LA for some finite algebra A?
![Page 11: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/11.jpg)
The homomorphism lattice induced by a finite algebra
FactLet A be a finite algebra. Then the variety Var(A) generated byA is locally finite.
NotationGiven a finite algebra A, we define the lattice
LA := 〈Var(A)fin/≡;→〉
and refer to it as the homomorphism lattice induced by A.
QuestionI Which lattices arise as LA for some finite algebra A?
![Page 12: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/12.jpg)
Warm-up exercisesExample 1. Representing finite chains
I The 2-element chain:
A = 〈{0,1}; f 〉
0 1f
LA ∼= 2
I The 3-element chain:
A = 〈{0,1,2,3}; f 〉
0 1 2 3f
LA ∼= 3
I In general, if A = 〈{0,1, . . . ,2n−1 − 1}; f 〉, where f issuccessor modulo 2n−1, then LA ∼= n.
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Warm-up exercisesExample 1. Representing finite chains
I The 2-element chain:
A = 〈{0,1}; f 〉
0 1f
LA ∼= 2
I The 3-element chain:
A = 〈{0,1,2,3}; f 〉
0 1 2 3f
LA ∼= 3
I In general, if A = 〈{0,1, . . . ,2n−1 − 1}; f 〉, where f issuccessor modulo 2n−1, then LA ∼= n.
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Warm-up exercisesExample 1. Representing finite chains
I The 2-element chain:
A = 〈{0,1}; f 〉
0 1f
LA ∼= 2
I The 3-element chain:
A = 〈{0,1,2,3}; f 〉
0 1 2 3f
LA ∼= 3
I In general, if A = 〈{0,1, . . . ,2n−1 − 1}; f 〉, where f issuccessor modulo 2n−1, then LA ∼= n.
![Page 15: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/15.jpg)
An interlude on cores
I A finite algebra C is a core if every endomorphism of C isan automorphism.
I For each finite algebra B, there is a subalgebra C (uniqueup to isomorphism) such that C is a core and there is aretraction ϕ : B→ C.
I If we let K consist of one copy (up to isomorphism) of eachcore in Var(A)fin, then K is a transversal of the equivalenceclasses of LA.
I Moreover, the cores are precisely the algebras that areminimal-sized within their equivalence classes.
I So the lattice LA is finite if and only if there is a finite boundon the sizes of the cores in Var(A)fin.
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An interlude on cores
I A finite algebra C is a core if every endomorphism of C isan automorphism.
I For each finite algebra B, there is a subalgebra C (uniqueup to isomorphism) such that C is a core and there is aretraction ϕ : B→ C.
I If we let K consist of one copy (up to isomorphism) of eachcore in Var(A)fin, then K is a transversal of the equivalenceclasses of LA.
I Moreover, the cores are precisely the algebras that areminimal-sized within their equivalence classes.
I So the lattice LA is finite if and only if there is a finite boundon the sizes of the cores in Var(A)fin.
![Page 17: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/17.jpg)
An interlude on cores
I A finite algebra C is a core if every endomorphism of C isan automorphism.
I For each finite algebra B, there is a subalgebra C (uniqueup to isomorphism) such that C is a core and there is aretraction ϕ : B→ C.
I If we let K consist of one copy (up to isomorphism) of eachcore in Var(A)fin, then K is a transversal of the equivalenceclasses of LA.
I Moreover, the cores are precisely the algebras that areminimal-sized within their equivalence classes.
I So the lattice LA is finite if and only if there is a finite boundon the sizes of the cores in Var(A)fin.
![Page 18: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/18.jpg)
An interlude on cores
I A finite algebra C is a core if every endomorphism of C isan automorphism.
I For each finite algebra B, there is a subalgebra C (uniqueup to isomorphism) such that C is a core and there is aretraction ϕ : B→ C.
I If we let K consist of one copy (up to isomorphism) of eachcore in Var(A)fin, then K is a transversal of the equivalenceclasses of LA.
I Moreover, the cores are precisely the algebras that areminimal-sized within their equivalence classes.
I So the lattice LA is finite if and only if there is a finite boundon the sizes of the cores in Var(A)fin.
![Page 19: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/19.jpg)
An interlude on cores
I A finite algebra C is a core if every endomorphism of C isan automorphism.
I For each finite algebra B, there is a subalgebra C (uniqueup to isomorphism) such that C is a core and there is aretraction ϕ : B→ C.
I If we let K consist of one copy (up to isomorphism) of eachcore in Var(A)fin, then K is a transversal of the equivalenceclasses of LA.
I Moreover, the cores are precisely the algebras that areminimal-sized within their equivalence classes.
I So the lattice LA is finite if and only if there is a finite boundon the sizes of the cores in Var(A)fin.
![Page 20: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/20.jpg)
Warm-up exercisesExample 2. The homomorphism lattice induced by a finite monounary algebra
Questions1. Which lattices can be obtained as LA, for some finite
monounary algebra A?
2. Can we obtain the lattice 22 in this way?
LemmaLet A = 〈A; f 〉 be a finite monounary algebra. Then, for some `and some k1, . . . , k`,
LA ∼= (k1 t · · · t k`)⊕ 1,
where the coproduct is in the variety D of distributive lattices.(Let n be the least common multiple of the sizes of the cyclesof A. Then pk1
1 · · · pk`` is the prime decomposition of n.)
I So the answer to Question 2 is ‘No’.
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Warm-up exercisesExample 2. The homomorphism lattice induced by a finite monounary algebra
Questions1. Which lattices can be obtained as LA, for some finite
monounary algebra A?2. Can we obtain the lattice 22 in this way?
LemmaLet A = 〈A; f 〉 be a finite monounary algebra. Then, for some `and some k1, . . . , k`,
LA ∼= (k1 t · · · t k`)⊕ 1,
where the coproduct is in the variety D of distributive lattices.(Let n be the least common multiple of the sizes of the cyclesof A. Then pk1
1 · · · pk`` is the prime decomposition of n.)
I So the answer to Question 2 is ‘No’.
![Page 22: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/22.jpg)
Warm-up exercisesExample 2. The homomorphism lattice induced by a finite monounary algebra
Questions1. Which lattices can be obtained as LA, for some finite
monounary algebra A?2. Can we obtain the lattice 22 in this way?
LemmaLet A = 〈A; f 〉 be a finite monounary algebra. Then, for some `and some k1, . . . , k`,
LA ∼= (k1 t · · · t k`)⊕ 1,
where the coproduct is in the variety D of distributive lattices.(Let n be the least common multiple of the sizes of the cyclesof A. Then pk1
1 · · · pk`` is the prime decomposition of n.)
I So the answer to Question 2 is ‘No’.
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Warm-up exercisesExample 2. The homomorphism lattice induced by a finite monounary algebra
Questions1. Which lattices can be obtained as LA, for some finite
monounary algebra A?2. Can we obtain the lattice 22 in this way?
LemmaLet A = 〈A; f 〉 be a finite monounary algebra. Then, for some `and some k1, . . . , k`,
LA ∼= (k1 t · · · t k`)⊕ 1,
where the coproduct is in the variety D of distributive lattices.(Let n be the least common multiple of the sizes of the cyclesof A. Then pk1
1 · · · pk`` is the prime decomposition of n.)
I So the answer to Question 2 is ‘No’.
![Page 24: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/24.jpg)
Warm-up exercisesExample 3. Representing the lattice 22
0
a b
1
A = 〈{0,a,b,1};∨,∧,0,1,a,b〉
a b
a
b a
b
a = b
LA ∼= 22
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Warm-up exercisesExample 4. Representing finite partition lattices
I To represent the lattice M3, we can use the nullary algebraA = 〈{1,2,3};1,2,3〉:
2=31=2 3=1
1=2=3
LA ∼= M3
I If A = 〈{1,2, . . . ,n};1,2, . . . ,n〉, then LA ∼= Equiv(n).
QuestionsI Can we represent each finite distributive lattice as LA?I We have represented M3. What about N5?
We will answer both of these questions later in the talk.
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Warm-up exercisesExample 4. Representing finite partition lattices
I To represent the lattice M3, we can use the nullary algebraA = 〈{1,2,3};1,2,3〉:
2=31=2 3=1
1=2=3
LA ∼= M3
I If A = 〈{1,2, . . . ,n};1,2, . . . ,n〉, then LA ∼= Equiv(n).
QuestionsI Can we represent each finite distributive lattice as LA?I We have represented M3. What about N5?
We will answer both of these questions later in the talk.
![Page 27: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/27.jpg)
Warm-up exercisesExample 4. Representing finite partition lattices
I To represent the lattice M3, we can use the nullary algebraA = 〈{1,2,3};1,2,3〉:
2=31=2 3=1
1=2=3
LA ∼= M3
I If A = 〈{1,2, . . . ,n};1,2, . . . ,n〉, then LA ∼= Equiv(n).
QuestionsI Can we represent each finite distributive lattice as LA?I We have represented M3. What about N5?
We will answer both of these questions later in the talk.
![Page 28: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/28.jpg)
How to find complicated examplesFinite-to-finite universal varieties
I A variety V of algebras is finite-to-finite universal if thecategory G of digraphs has a finiteness-preserving fullembedding into V.
I This is equivalent to requiring that every variety of algebrashas a finiteness-preserving full embedding into V.
I If Var(A) is finite-to-finite universal, thenI there is an order-embedding of PGfin into LA,I every countable ordered set order-embeds into LA,I in particular, LB order-embeds into LA, for all finite B.
![Page 29: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/29.jpg)
How to find complicated examplesFinite-to-finite universal varieties
I A variety V of algebras is finite-to-finite universal if thecategory G of digraphs has a finiteness-preserving fullembedding into V.
I This is equivalent to requiring that every variety of algebrashas a finiteness-preserving full embedding into V.
I If Var(A) is finite-to-finite universal, thenI there is an order-embedding of PGfin into LA,I every countable ordered set order-embeds into LA,I in particular, LB order-embeds into LA, for all finite B.
![Page 30: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/30.jpg)
How to find complicated examplesFinite-to-finite universal varieties
I A variety V of algebras is finite-to-finite universal if thecategory G of digraphs has a finiteness-preserving fullembedding into V.
I This is equivalent to requiring that every variety of algebrashas a finiteness-preserving full embedding into V.
I If Var(A) is finite-to-finite universal, thenI there is an order-embedding of PGfin into LA,I every countable ordered set order-embeds into LA,I in particular, LB order-embeds into LA, for all finite B.
![Page 31: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/31.jpg)
Finite-to-finite universal varieties
Finite algebras A with Var(A) finite-to-finite universal:
1. The bounded lattice M3. [Goralcík, Koubek, Sichler 1990]
2. The algebra A = 〈A;∨,∧,0,1,a,b〉, where〈A;∨,∧,0,1〉 is the bounded distributive lattice 1⊕ 22 ⊕ 1freely generated by {a,b}.[Uses Adams, Koubek, Sichler 1985]
3. Let P = 〈{0,1,2}; ·〉 be the semigroupshown on the right.
Let A = 〈A; ·,a,b, c〉, where 〈A; ·〉 isfreely generated in Var(P)by {u, v ,a,b, c}.[Uses Demlová, Koubek 2003]
· 0 1 20 0 0 01 0 1 22 0 0 0
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Finite-to-finite universal varieties
Finite algebras A with Var(A) finite-to-finite universal:
1. The bounded lattice M3. [Goralcík, Koubek, Sichler 1990]
2. The algebra A = 〈A;∨,∧,0,1,a,b〉, where〈A;∨,∧,0,1〉 is the bounded distributive lattice 1⊕ 22 ⊕ 1freely generated by {a,b}.[Uses Adams, Koubek, Sichler 1985]
3. Let P = 〈{0,1,2}; ·〉 be the semigroupshown on the right.
Let A = 〈A; ·,a,b, c〉, where 〈A; ·〉 isfreely generated in Var(P)by {u, v ,a,b, c}.[Uses Demlová, Koubek 2003]
· 0 1 20 0 0 01 0 1 22 0 0 0
![Page 33: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/33.jpg)
Finite-to-finite universal varieties
Finite algebras A with Var(A) finite-to-finite universal:
1. The bounded lattice M3. [Goralcík, Koubek, Sichler 1990]
2. The algebra A = 〈A;∨,∧,0,1,a,b〉, where〈A;∨,∧,0,1〉 is the bounded distributive lattice 1⊕ 22 ⊕ 1freely generated by {a,b}.[Uses Adams, Koubek, Sichler 1985]
3. Let P = 〈{0,1,2}; ·〉 be the semigroupshown on the right.
Let A = 〈A; ·,a,b, c〉, where 〈A; ·〉 isfreely generated in Var(P)by {u, v ,a,b, c}.[Uses Demlová, Koubek 2003]
· 0 1 20 0 0 01 0 1 22 0 0 0
![Page 34: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/34.jpg)
Finite-to-finite universal varieties
Finite algebras A with Var(A) finite-to-finite universal:
1. The bounded lattice M3. [Goralcík, Koubek, Sichler 1990]
2. The algebra A = 〈A;∨,∧,0,1,a,b〉, where〈A;∨,∧,0,1〉 is the bounded distributive lattice 1⊕ 22 ⊕ 1freely generated by {a,b}.[Uses Adams, Koubek, Sichler 1985]
3. Let P = 〈{0,1,2}; ·〉 be the semigroupshown on the right.
Let A = 〈A; ·,a,b, c〉, where 〈A; ·〉 isfreely generated in Var(P)by {u, v ,a,b, c}.[Uses Demlová, Koubek 2003]
· 0 1 20 0 0 01 0 1 22 0 0 0
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Finite-to-finite universal varieties
Even a small unary algebra can generate a finite-to-finiteuniversal variety.
4. Let A = 〈{0,1,2,u, v}; f1, f2〉 be the 5-element unaryalgebra shown below.
1 2
u v
0
A
f2f1
a b c
G
a b c
u v
(a,b)(a,a) (b, c) (c,b)
G∗
The map G 7→ G∗ embeds the category G of digraphs intothe category Var(A). [Uses Hedrlín, Pultr 1966]
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Distributive lattices via quasi-primal algebras
A finite algebra A is quasi-primal if the ternary discriminatoroperation τ is a term function of A, where
τ(x , y , z) :=
{x if x 6= y ,z if x = y .
Theorem (Pixley 1970)A finite algebra A is quasi-primal if and only if every non-trivialsubalgebra of A is simple and the variety Var(A) is bothcongruence permutable and congruence distributive.
Theorem (Pixley 1970)Let A be a quasi-primal algebra. Then every finite algebra inVar(A) is isomorphic to a product of subalgebras of A.
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Distributive lattices via quasi-primal algebras
NotationI We use Sub(A) to denote the set of all subalgebras of an
algebra A. Note that 〈Sub(A);6〉 is a join-semilattice.I For an ordered set P, we use O(P) to denote the lattice of
all down-sets of P, ordered by inclusion.
Theorem (Birkhoff 1933)For every finite distributive lattice L, there exists a finite orderedset P such that L ∼= O(P). Indeed,
L ∼= O(J (L)),
where J (L) is the ordered set of join-irreducible elements of L.
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Distributive lattices via quasi-primal algebras
TheoremLet A be a quasi-primal algebra. Define the ordered set
Q := 〈Sub(A)/≡;→〉
and let Q denote Q without its top.a. If A has no trivial subalgebras, then LA ∼= O(Q).b. If A has a trivial subalgebra, then LA ∼= O(Q).
This result gives a strategy to show that every finite distributivelattice L arises as LA, for some finite algebra A:
I Let P = J (L). Hence, O(P) ∼= L.I Let P> denote P with a new top element > added.I Construct a quasi-primal algebra A, with a trivial
subalgebra, such that 〈Sub(A)/≡;→〉 ∼= P>. Then
LA ∼= O(P>) = O(P) ∼= L.
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Distributive lattices via quasi-primal algebras
TheoremLet A be a quasi-primal algebra. Define the ordered set
Q := 〈Sub(A)/≡;→〉
and let Q denote Q without its top.a. If A has no trivial subalgebras, then LA ∼= O(Q).b. If A has a trivial subalgebra, then LA ∼= O(Q).
This result gives a strategy to show that every finite distributivelattice L arises as LA, for some finite algebra A:
I Let P = J (L). Hence, O(P) ∼= L.I Let P> denote P with a new top element > added.I Construct a quasi-primal algebra A, with a trivial
subalgebra, such that 〈Sub(A)/≡;→〉 ∼= P>. Then
LA ∼= O(P>) = O(P) ∼= L.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.I Let ↓x denote the subalgebra of A with universe ↓x .I Then α : S→ Sub(A), given by α(x) := ↓x , is an
order-isomorphism.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.I Let ↓x denote the subalgebra of A with universe ↓x .I Then α : S→ Sub(A), given by α(x) := ↓x , is an
order-isomorphism.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.I Let ↓x denote the subalgebra of A with universe ↓x .I Then α : S→ Sub(A), given by α(x) := ↓x , is an
order-isomorphism.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.
I Let ↓x denote the subalgebra of A with universe ↓x .I Then α : S→ Sub(A), given by α(x) := ↓x , is an
order-isomorphism.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.I Let ↓x denote the subalgebra of A with universe ↓x .
I Then α : S→ Sub(A), given by α(x) := ↓x , is anorder-isomorphism.
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Interlude: a motivating example
Lemma (Birkhoff, Frink 1948)For each finite join-semilattice S, there exists a finite algebra Asuch that 〈Sub(A);6〉 is order-isomorphic to S.
Proof.Let S = 〈S;∨〉 be a finite semilattice.
I For each covering pair a ≺ b in S,define fba : S → S by
fba(x) :=
{a if x = b,x otherwise.
b
a
x 6= bfba
I Now define F := { fba | a ≺ b in S } and A := 〈S;∨,F 〉.I Let ↓x denote the subalgebra of A with universe ↓x .I Then α : S→ Sub(A), given by α(x) := ↓x , is an
order-isomorphism.
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From L to the quasi-primal algebra A
5 6
4
3
1 2
L
5 6
3 4
1 2
P
5 6
3 4
1 2
0 = >
P>
1. Let L be a finite distributive lattice.
2. Let P = J (L).3. Let P> denote P with a new top element > added.
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From L to the quasi-primal algebra A
5 6
4
3
1 2
L
5 6
3 4
1 2
P
5 6
3 4
1 2
0 = >
P>
1. Let L be a finite distributive lattice.
2. Let P = J (L).3. Let P> denote P with a new top element > added.
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From L to the quasi-primal algebra A
5 6
4
3
1 2
L
5 6
3 4
1 2
P
5 6
3 4
1 2
0 = >
P>
1. Let L be a finite distributive lattice.2. Let P = J (L).
3. Let P> denote P with a new top element > added.
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From L to the quasi-primal algebra A
5 6
4
3
1 2
L
5 6
3 4
1 2
P
5 6
3 4
1 2
0 = >
P>
1. Let L be a finite distributive lattice.2. Let P = J (L).3. Let P> denote P with a new top element > added.
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From L to the quasi-primal algebra A
4. Let S be the universal covering tree of the ordered set P>.
5 6
3 4
1 2
0 = >
P>
5310 5320 6310 6320 6410 6420
310 320 410 420
10 20
0
S
ϕ
5. As a first approximation to A, let A1 = 〈S;∨,F1, τ〉, whereF1 := { fba | a ≺ b in S } and τ is the ternary discriminator.
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From L to the quasi-primal algebra A
5. A1 = 〈S;∨,F1, τ〉, where F1 := { fba | a ≺ b in S } and τ isthe ternary discriminator.
P>
b
a
S
fba
ϕ
Bad: We have ↓x 6∼= ↓y for all x 6= y in S\Min(S),but want ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).
Fix: Redefine fba and index via covers a ≺ b in P>.
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From L to the quasi-primal algebra A
5. A1 = 〈S;∨,F1, τ〉, where F1 := { fba | a ≺ b in S } and τ isthe ternary discriminator.
P>
b
a
S
fba
ϕ
Bad: We have ↓x 6∼= ↓y for all x 6= y in S\Min(S),but want ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).
Fix: Redefine fba and index via covers a ≺ b in P>.
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From L to the quasi-primal algebra A
5. A1 = 〈S;∨,F1, τ〉, where F1 := { fba | a ≺ b in S } and τ isthe ternary discriminator.
P>
b
aS
fba
ϕ
Bad: We have ↓x 6∼= ↓y for all x 6= y in S\Min(S),but want ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).
Fix: Redefine fba and index via covers a ≺ b in P>.
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From L to the quasi-primal algebra A
6. A2 = 〈S;∨,F2, τ〉, where F2 := { fba | a ≺ b in P> } and τ isthe ternary discriminator.
P>
m
S
S]gm
ϕ
Good: We now have ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).Bad: All subalgebras of A2 are homomorphically equivalent.
Fix: Let S] be S with every minimal element doubled.Then add operations gm, indexed by Min(P>), that flip thenew and old minimal elements in an appropriate way.
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From L to the quasi-primal algebra A
6. A2 = 〈S;∨,F2, τ〉, where F2 := { fba | a ≺ b in P> } and τ isthe ternary discriminator.
P>
mS
S]
gm
ϕ
Good: We now have ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).Bad: All subalgebras of A2 are homomorphically equivalent.Fix: Let S] be S with every minimal element doubled.
Then add operations gm, indexed by Min(P>), that flip thenew and old minimal elements in an appropriate way.
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From L to the quasi-primal algebra A
6. A2 = 〈S;∨,F2, τ〉, where F2 := { fba | a ≺ b in P> } and τ isthe ternary discriminator.
P>m
S
S]gm
ϕ
Good: We now have ↓x ∼= ↓y whenever ϕ(x) = ϕ(y).Bad: All subalgebras of A2 are homomorphically equivalent.Fix: Let S] be S with every minimal element doubled.
Then add operations gm, indexed by Min(P>), that flip thenew and old minimal elements in an appropriate way.
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From L to the quasi-primal algebra A
7. A3 = 〈S];∨,F2,G, τ〉, where F2 := { fba | a ≺ b in P> },G = {gm | m ∈ Min(P>) }, τ is the ternary discriminator.
P> S]
ϕ
Good: Now every subalgebra of A3 is of the form ↓x , for somex ∈ S, and ↓x ≡ ↓y ⇐⇒ ↓x ∼= ↓y ⇐⇒ ϕ(x) = ϕ(y).
Bad: A3 has no trivial subalgebras.Fix: Remove all f>a from F2
and add an operation h so that(∗) {>} is the only new subuniverse.
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From L to the quasi-primal algebra A
7. A3 = 〈S];∨,F2,G, τ〉, where F2 := { fba | a ≺ b in P> },G = {gm | m ∈ Min(P>) }, τ is the ternary discriminator.
P> S]
ϕ
Good: Now every subalgebra of A3 is of the form ↓x , for somex ∈ S, and ↓x ≡ ↓y ⇐⇒ ↓x ∼= ↓y ⇐⇒ ϕ(x) = ϕ(y).
Bad: A3 has no trivial subalgebras.
Fix: Remove all f>a from F2
and add an operation h so that(∗) {>} is the only new subuniverse.
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From L to the quasi-primal algebra A
7. A3 = 〈S];∨,F2,G, τ〉, where F2 := { fba | a ≺ b in P> },G = {gm | m ∈ Min(P>) }, τ is the ternary discriminator.
P> S]
ϕ
Good: Now every subalgebra of A3 is of the form ↓x , for somex ∈ S, and ↓x ≡ ↓y ⇐⇒ ↓x ∼= ↓y ⇐⇒ ϕ(x) = ϕ(y).
Bad: A3 has no trivial subalgebras.Fix: Remove all f>a from F2
and add an operation h so that(∗) {>} is the only new subuniverse.
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From L to the quasi-primal algebra A
7. A3 = 〈S];∨,F2,G, τ〉, where F2 := { fba | a ≺ b in P> },G = {gm | m ∈ Min(P>) }, τ is the ternary discriminator.
P> S]
ϕ
Good: Now every subalgebra of A3 is of the form ↓x , for somex ∈ S, and ↓x ≡ ↓y ⇐⇒ ↓x ∼= ↓y ⇐⇒ ϕ(x) = ϕ(y).
Bad: A3 has no trivial subalgebras.Fix: Remove all f>a from F2 and add an operation h so that
(∗) {>} is the only new subuniverse.
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From L to the quasi-primal algebra A
8. A = 〈S];∨,F3,G,h, τ〉, where F3 := { fba | a ≺ b in P },G = {gm | m ∈ Min(P>) }, h has property (∗), and τ is theternary discriminator.
P> S]
ϕ
Now every non-trivial subalgebra of A is of the form ↓x , forsome x ∈ S, and ↓x → ↓y if and only if ϕ(x) 6 ϕ(y).Thus A is a quasi-primal algebra, with a trivial subalgebra, suchthat 〈Sub(A)/≡;→〉 ∼= P>. Hence LA ∼= O(P>) = O(P) ∼= L.
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From L to the quasi-primal algebra A
8. A = 〈S];∨,F3,G,h, τ〉, where F3 := { fba | a ≺ b in P },G = {gm | m ∈ Min(P>) }, h has property (∗), and τ is theternary discriminator.
P> S]
ϕ
Now every non-trivial subalgebra of A is of the form ↓x , forsome x ∈ S, and ↓x → ↓y if and only if ϕ(x) 6 ϕ(y).
Thus A is a quasi-primal algebra, with a trivial subalgebra, suchthat 〈Sub(A)/≡;→〉 ∼= P>. Hence LA ∼= O(P>) = O(P) ∼= L.
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From L to the quasi-primal algebra A
8. A = 〈S];∨,F3,G,h, τ〉, where F3 := { fba | a ≺ b in P },G = {gm | m ∈ Min(P>) }, h has property (∗), and τ is theternary discriminator.
P> S]
ϕ
Now every non-trivial subalgebra of A is of the form ↓x , forsome x ∈ S, and ↓x → ↓y if and only if ϕ(x) 6 ϕ(y).Thus A is a quasi-primal algebra, with a trivial subalgebra, suchthat 〈Sub(A)/≡;→〉 ∼= P>. Hence LA ∼= O(P>) = O(P) ∼= L.
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From L to the quasi-primal algebra A
L A
TheoremFor each finite distributive lattice L, there exists a quasi-primalalgebra A such that LA is isomorphic to L.
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The class Lhom
NotationLet Lhom be the class of all lattices L such that L ∼= LA, for somefinite algebra A.
QuestionsI We wish to know which (finite) lattices belong to Lhom.I What general properties does the class Lhom have?
For example:I We know Lhom is closed under finite products.I It is not clear whether Lhom is closed under forming
homomorphic images or taking sublattices.
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Congruence lattices
Finite Lattice Representation ProblemDoes every finite lattice arises as the congruence lattice of afinite algebra?
I This is one of the most famous unsolved problems inuniversal algebra.
I It is therefore natural to ask whether the congruence latticeof each finite algebra belongs to Lhom.
I That is, given a finite algebra A, does there exist a finitealgebra B such that LB ∼= Con(A)?
I We focus on the special case where LA ∼= Con(A).
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Congruence lattices
I We can assume, without affecting the lattice Con(A), thatevery element of A is the value of a nullary term function.
I Then, for all B ∈ Var(A), the values of the nullary termfunctions of B form a subalgebra that we will denote by B0.
Congruence Lattice LemmaLet A be a finite algebra such that each element is the value ofa nullary term function. Then the following are equivalent:
1. LA ∼= Con(A);2. for every finite algebra B ∈ Var(A), we have B→ B0;3. a. B→ B0, for every finite subdirectly irreducible algebra
B ∈ Var(A), andb. (A/θ1)× (A/θ2)→ A/(θ1 ∩ θ2), for all θ1, θ2 ∈ Con(A).
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Congruence lattices
I We can assume, without affecting the lattice Con(A), thatevery element of A is the value of a nullary term function.
I Then, for all B ∈ Var(A), the values of the nullary termfunctions of B form a subalgebra that we will denote by B0.
Congruence Lattice LemmaLet A be a finite algebra such that each element is the value ofa nullary term function. Then the following are equivalent:
1. LA ∼= Con(A);2. for every finite algebra B ∈ Var(A), we have B→ B0;3. a. B→ B0, for every finite subdirectly irreducible algebra
B ∈ Var(A), andb. (A/θ1)× (A/θ2)→ A/(θ1 ∩ θ2), for all θ1, θ2 ∈ Con(A).
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Subspace lattices and partition lattices
Corollary of (2)⇒ (1)Let A be a finite algebra.
I Assume that every finite algebra in Var(A) has a retractiononto each of its subalgebras.
I Let A+ be the algebra obtained from A by making eachelement of A the value of a nullary operation.
Then LA+ ∼= Con(A), whence Con(A) ∈ Lhom.
Applications of the corollaryI Let V be a finite dimensional vector space over a finite
field. The lattice of subspaces of V belongs to Lhom.I Let S be a finite set. The lattice of equivalences on S
belongs to Lhom. That is, the class Lhom contains all finitepartition lattices.
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Subspace lattices and partition lattices
Corollary of (2)⇒ (1)Let A be a finite algebra.
I Assume that every finite algebra in Var(A) has a retractiononto each of its subalgebras.
I Let A+ be the algebra obtained from A by making eachelement of A the value of a nullary operation.
Then LA+ ∼= Con(A), whence Con(A) ∈ Lhom.
Applications of the corollaryI Let V be a finite dimensional vector space over a finite
field. The lattice of subspaces of V belongs to Lhom.
I Let S be a finite set. The lattice of equivalences on Sbelongs to Lhom. That is, the class Lhom contains all finitepartition lattices.
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Subspace lattices and partition lattices
Corollary of (2)⇒ (1)Let A be a finite algebra.
I Assume that every finite algebra in Var(A) has a retractiononto each of its subalgebras.
I Let A+ be the algebra obtained from A by making eachelement of A the value of a nullary operation.
Then LA+ ∼= Con(A), whence Con(A) ∈ Lhom.
Applications of the corollaryI Let V be a finite dimensional vector space over a finite
field. The lattice of subspaces of V belongs to Lhom.I Let S be a finite set. The lattice of equivalences on S
belongs to Lhom. That is, the class Lhom contains all finitepartition lattices.
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Partition lattices
Theorem (Pudlák, Tuma 1980)Every finite lattice embeds into a finite partition lattice.
ConsequencesI Every finite lattice embeds into a finite lattice in Lhom.I Thus, if we could show that Lhom were closed under
forming sublattices, then it would follow that every finitelattice belongs to Lhom.
I Since the variety of lattices is generated by its finitemembers, the only equations true in the class Lhom arethose true in all lattices.
I In fact, more is true. The only universal first-ordersentences true in the class Lhom are those true in alllattices.
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Partition lattices
Theorem (Pudlák, Tuma 1980)Every finite lattice embeds into a finite partition lattice.
ConsequencesI Every finite lattice embeds into a finite lattice in Lhom.I Thus, if we could show that Lhom were closed under
forming sublattices, then it would follow that every finitelattice belongs to Lhom.
I Since the variety of lattices is generated by its finitemembers, the only equations true in the class Lhom arethose true in all lattices.
I In fact, more is true. The only universal first-ordersentences true in the class Lhom are those true in alllattices.
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Partition lattices
Theorem (Pudlák, Tuma 1980)Every finite lattice embeds into a finite partition lattice.
ConsequencesI Every finite lattice embeds into a finite lattice in Lhom.I Thus, if we could show that Lhom were closed under
forming sublattices, then it would follow that every finitelattice belongs to Lhom.
I Since the variety of lattices is generated by its finitemembers, the only equations true in the class Lhom arethose true in all lattices.
I In fact, more is true. The only universal first-ordersentences true in the class Lhom are those true in alllattices.
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Partition lattices
Theorem (Pudlák, Tuma 1980)Every finite lattice embeds into a finite partition lattice.
ConsequencesI Every finite lattice embeds into a finite lattice in Lhom.I Thus, if we could show that Lhom were closed under
forming sublattices, then it would follow that every finitelattice belongs to Lhom.
I Since the variety of lattices is generated by its finitemembers, the only equations true in the class Lhom arethose true in all lattices.
I In fact, more is true. The only universal first-ordersentences true in the class Lhom are those true in alllattices.
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Subgroup lattices
The following are equivalent [Pálfy, Pudlák 1980]:
I every finite lattice arises as the congruence lattice of afinite algebra;
I every finite lattice arises as an interval [H,G] in thesubgroup lattice of a finite group G.
So it is natural to investigate which intervals in subgroup latticesbelong to Lhom.
Application of the lemma (Keith Kearnes)I Let G be a group and let A = 〈A; {λg | g ∈ G}〉 be a G-set,
with at least one singleton orbit, regarded as a unaryalgebra. Then LA+ ∼= Con(A).
I If A = G/H ∪ {∞} with the natural left action of G oncosets and with∞ fixed, then LA+ ∼= Con(A) ∼= [H,G]⊕ 1.
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Subgroup lattices
The following are equivalent [Pálfy, Pudlák 1980]:
I every finite lattice arises as the congruence lattice of afinite algebra;
I every finite lattice arises as an interval [H,G] in thesubgroup lattice of a finite group G.
So it is natural to investigate which intervals in subgroup latticesbelong to Lhom.
Application of the lemma (Keith Kearnes)I Let G be a group and let A = 〈A; {λg | g ∈ G}〉 be a G-set,
with at least one singleton orbit, regarded as a unaryalgebra. Then LA+ ∼= Con(A).
I If A = G/H ∪ {∞} with the natural left action of G oncosets and with∞ fixed, then LA+ ∼= Con(A) ∼= [H,G]⊕ 1.
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The lattice N5
I With the exception of N5, we now know that all lattices ofsize up to 5 belong to Lhom.
I Our aim now is to give an example of a finite algebra Asuch that LA ∼= N5.
I We shall apply (3)⇒ (1) of the Congruence LatticeLemma.
Congruence Lattice Lemma
1. LA ∼= Con(A);3. a. B→ B0, for every finite subdirectly irreducible algebra
B ∈ Var(A), andb. (A/θ1)× (A/θ2)→ A/(θ1 ∩ θ2), for all θ1, θ2 ∈ Con(A).
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The lattice N5
I With the exception of N5, we now know that all lattices ofsize up to 5 belong to Lhom.
I Our aim now is to give an example of a finite algebra Asuch that LA ∼= N5.
I We shall apply (3)⇒ (1) of the Congruence LatticeLemma.
Congruence Lattice Lemma
1. LA ∼= Con(A);3. a. B→ B0, for every finite subdirectly irreducible algebra
B ∈ Var(A), andb. (A/θ1)× (A/θ2)→ A/(θ1 ∩ θ2), for all θ1, θ2 ∈ Con(A).
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Distributive bisemilattices
I An algebra A = 〈A;∧,u〉 is a distributive bisemilattice ifboth ∧ and u are semilattice operations on A and eachdistributes over the other.
I Up to isomorphism, the subdirectly irreducible distributivebisemilattices are D, S and L. [Kalman 1971]
I S is term equivalent to the two-element semilattice.I L is the two-element lattice.
x
y
z
∧
D x
z
y
u
x
y
∧
S x
y
u
y
z
∧
L z
y
u
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Distributive bisemilattices
I An algebra A = 〈A;∧,u〉 is a distributive bisemilattice ifboth ∧ and u are semilattice operations on A and eachdistributes over the other.
I Up to isomorphism, the subdirectly irreducible distributivebisemilattices are D, S and L. [Kalman 1971]
I S is term equivalent to the two-element semilattice.I L is the two-element lattice.
x
y
z
∧
D x
z
y
u
x
y
∧
S x
y
u
y
z
∧
L z
y
u
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Distributive bisemilattices
I An algebra A = 〈A;∧,u〉 is a distributive bisemilattice ifboth ∧ and u are semilattice operations on A and eachdistributes over the other.
I Up to isomorphism, the subdirectly irreducible distributivebisemilattices are D, S and L. [Kalman 1971]
I S is term equivalent to the two-element semilattice.
I L is the two-element lattice.
x
y
z
∧
D x
z
y
u
x
y
∧
S x
y
u
y
z
∧
L z
y
u
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Distributive bisemilattices
I An algebra A = 〈A;∧,u〉 is a distributive bisemilattice ifboth ∧ and u are semilattice operations on A and eachdistributes over the other.
I Up to isomorphism, the subdirectly irreducible distributivebisemilattices are D, S and L. [Kalman 1971]
I S is term equivalent to the two-element semilattice.I L is the two-element lattice.
x
y
z
∧
D x
z
y
u
x
y
∧
S x
y
u
y
z
∧
L z
y
u
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The lattice N5
ExampleLet A = 〈{0,a,b,1};∧,u,0,a,b,1〉 be the distributivebisemilattice S× L with all four elements added as nullaries.Then LA ∼= N5.
(x , y)
0
(y , y)
a
(x , z)
b
(y , z)
1
∧(x , z)
b
(x , y)
0
(y , z)
1
(y , y)
a
uThe distributive bisemilattice S× L
I The bisemilattice S× L is a reduct of the 4-elementdistributive bilattice made famous by Belnap (1977).
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The lattice N5
ExampleLet A = 〈{0,a,b,1};∧,u,0,a,b,1〉 be the distributivebisemilattice S× L with all four elements added as nullaries.Then LA ∼= N5.
(x , y)
0
(y , y)
a
(x , z)
b
(y , z)
1
∧
(x , z)
b
(x , y)
0
(y , z)
1
(y , y)
a
uThe algebra A
I The bisemilattice S× L is a reduct of the 4-elementdistributive bilattice made famous by Belnap (1977).
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The lattice N5
ExampleLet A = 〈{0,a,b,1};∧,u,0,a,b,1〉 be the distributivebisemilattice S× L with all four elements added as nullaries.Then LA ∼= N5.
(x , y)
0
(y , y)
a
(x , z)
b
(y , z)
1
∧
(x , z)
b
(x , y)
0
(y , z)
1
(y , y)
a
uThe algebra A
I The bisemilattice S× L is a reduct of the 4-elementdistributive bilattice made famous by Belnap (1977).
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Proof of the N5 example
I The algebra A:
0
a b
1
∧b
0 1
a
u
I The congruences on A:
0
a b
1
α
0
a b
1
β
0
a b
1
γ
0A
α
γβ
1A
Con(A)
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Proof of the N5 example
I The congruences on A:
0
a b
1
α
0
a b
1
β
0
a b
1
γ
0A
α
γβ
1A
Con(A)
I The corresponding quotients:
0b
a1
∧A/β 0b
a1
u0a
b1
∧A/γ b1
0a
u0b
a
1
∧A/α 0b
1
a
u
Skip proof
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Proof of the N5 example
0b
a1
∧A/β 0b
a1
u0a
b1
∧A/γ b1
0a
u0b
a
1
∧A/α 0b
1
a
u
I Since α, β and γ are meet-irreducible in Con(A), it followsthat A/α, A/β and A/γ are subdirectly irreducible algebrasin Var(A).
I We claim that, up to isomorphism, these are the onlysubdirectly irreducible algebras in Var(A).
I Note that each of these algebras satisfies B = B0.
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Proof of the N5 example
0b
a1
∧A/β 0b
a1
u0a
b1
∧A/γ b1
0a
u0b
a
1
∧A/α 0b
1
a
u
I Since α, β and γ are meet-irreducible in Con(A), it followsthat A/α, A/β and A/γ are subdirectly irreducible algebrasin Var(A).
I We claim that, up to isomorphism, these are the onlysubdirectly irreducible algebras in Var(A).
I Note that each of these algebras satisfies B = B0.
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Proof of the N5 example
0b
a1
∧A/β 0b
a1
u0a
b1
∧A/γ b1
0a
u0b
a
1
∧A/α 0b
1
a
u
I Since α, β and γ are meet-irreducible in Con(A), it followsthat A/α, A/β and A/γ are subdirectly irreducible algebrasin Var(A).
I We claim that, up to isomorphism, these are the onlysubdirectly irreducible algebras in Var(A).
I Note that each of these algebras satisfies B = B0.
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Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
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Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
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Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
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Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
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Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
![Page 97: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/97.jpg)
Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
![Page 98: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/98.jpg)
Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
![Page 99: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/99.jpg)
Proof of the N5 example
x
y
∧S x
y
uy
z
∧L z
y
ux
y
z
∧D x
z
y
u
0b
a1
∧0b
a1
u0a
b1
∧b1
0a
u0b
a
1
∧0b
1
a
u
I 0 is the bottom for ∧I 1 is the top for ∧I a is the top for uI b is the bottom for u 0
a b
1
∧b
0 1
a
u
![Page 100: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/100.jpg)
Proof of the N5 example
I Hence every subdirectly irreducible algebra in Var(A) isisomorphic to one of A/α, A/β and A/γ.
I So (3a) of the Congruence Lattice Lemma holds:i.e., every subdirectly irreducible algebra B in Var(A)satisfies B→ B0 (indeed, B = B0).
I We now check that (3b) holds:i.e., for all θ1, θ2 ∈ Con(A),
(A/θ1)×(A/θ2)→ A/(θ1∧θ2).0A
αγ
β1A
Con(A)
I Since α 6 β and β ∧ γ = 0A, it is enough to show that(A/β)× (A/γ) ∼= A, which follows immediately from theeasily checked fact that β ∧ γ = 0A and β · γ = 1A.
I Hence LA ∼= Con(A) ∼= N5, by the Congruence LatticeLemma.
![Page 101: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/101.jpg)
Proof of the N5 example
I Hence every subdirectly irreducible algebra in Var(A) isisomorphic to one of A/α, A/β and A/γ.
I So (3a) of the Congruence Lattice Lemma holds:i.e., every subdirectly irreducible algebra B in Var(A)satisfies B→ B0 (indeed, B = B0).
I We now check that (3b) holds:i.e., for all θ1, θ2 ∈ Con(A),
(A/θ1)×(A/θ2)→ A/(θ1∧θ2).0A
αγ
β1A
Con(A)
I Since α 6 β and β ∧ γ = 0A, it is enough to show that(A/β)× (A/γ) ∼= A, which follows immediately from theeasily checked fact that β ∧ γ = 0A and β · γ = 1A.
I Hence LA ∼= Con(A) ∼= N5, by the Congruence LatticeLemma.
![Page 102: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/102.jpg)
Proof of the N5 example
I Hence every subdirectly irreducible algebra in Var(A) isisomorphic to one of A/α, A/β and A/γ.
I So (3a) of the Congruence Lattice Lemma holds:i.e., every subdirectly irreducible algebra B in Var(A)satisfies B→ B0 (indeed, B = B0).
I We now check that (3b) holds:i.e., for all θ1, θ2 ∈ Con(A),
(A/θ1)×(A/θ2)→ A/(θ1∧θ2).0A
αγ
β1A
Con(A)
I Since α 6 β and β ∧ γ = 0A, it is enough to show that(A/β)× (A/γ) ∼= A, which follows immediately from theeasily checked fact that β ∧ γ = 0A and β · γ = 1A.
I Hence LA ∼= Con(A) ∼= N5, by the Congruence LatticeLemma.
![Page 103: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/103.jpg)
Proof of the N5 example
I Hence every subdirectly irreducible algebra in Var(A) isisomorphic to one of A/α, A/β and A/γ.
I So (3a) of the Congruence Lattice Lemma holds:i.e., every subdirectly irreducible algebra B in Var(A)satisfies B→ B0 (indeed, B = B0).
I We now check that (3b) holds:i.e., for all θ1, θ2 ∈ Con(A),
(A/θ1)×(A/θ2)→ A/(θ1∧θ2).0A
αγ
β1A
Con(A)
I Since α 6 β and β ∧ γ = 0A, it is enough to show that(A/β)× (A/γ) ∼= A, which follows immediately from theeasily checked fact that β ∧ γ = 0A and β · γ = 1A.
I Hence LA ∼= Con(A) ∼= N5, by the Congruence LatticeLemma.
![Page 104: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/104.jpg)
Proof of the N5 example
I Hence every subdirectly irreducible algebra in Var(A) isisomorphic to one of A/α, A/β and A/γ.
I So (3a) of the Congruence Lattice Lemma holds:i.e., every subdirectly irreducible algebra B in Var(A)satisfies B→ B0 (indeed, B = B0).
I We now check that (3b) holds:i.e., for all θ1, θ2 ∈ Con(A),
(A/θ1)×(A/θ2)→ A/(θ1∧θ2).0A
αγ
β1A
Con(A)
I Since α 6 β and β ∧ γ = 0A, it is enough to show that(A/β)× (A/γ) ∼= A, which follows immediately from theeasily checked fact that β ∧ γ = 0A and β · γ = 1A.
I Hence LA ∼= Con(A) ∼= N5, by the Congruence LatticeLemma.
![Page 105: The homomorphism lattice induced by a finite algebra...Some examples I The case where C is the category of finite graphs or finite digraphs has been studied extensively: I in both](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc3c855e5e9f717ef140748/html5/thumbnails/105.jpg)
ProblemsRepresentation
I Does every countable bounded lattice belong to Lhom?I In particular, does Lhom contain:
I every finite lattice?I the congruence lattice of every finite algebra?I every interval in the subgroup lattice of a finite group?
I Is every finite homomorphism lattice LA representable asthe congruence lattice of a finite algebra?
FinitenessI For which finite algebras A is the lattice LA finite?
Is this decidable?
Classes of algebrasI Restrict to natural classes of algebras. For example:
unary algebras, semigroups with constants.