On the relationship of maximal clones and maximal C-clones Mike - - PowerPoint PPT Presentation

Galois theory C-clones Max clones

On the relationship of maximal clones and maximal C-clones

Mike Behrisch

Institute of Computer Languages, Theory and Logic Group, Vienna University of Technology

21st June 2014 Warsaw, Poland

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Outline

1

Galois theory for clones

2

Clausal relations and clausal clones

3

Maximal clones/C-clones

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Presenting joint work with. . .

Edith Vargas-García Universidad Autónoma de la Ciudad de México Academia de Matemáticas

&&

University of Leeds School of Mathematics.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Outline

1

Galois theory for clones

2

Clausal relations and clausal clones

3

Maximal clones/C-clones

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Galois theory for clones

Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Galois theory for clones

Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D Finitary operations For k ∈ N+ any f : Dk − → D is a k-ary operation on D O(k)

D := DDk set of k-ary operations on D

OD :=

k∈N+ O(k) D

set of all finitary operations on D

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Galois theory for clones

Basis Galois correspondence connecting finitary operations on D ← → finitary relations on D Finitary operations For k ∈ N+ any f : Dk − → D is a k-ary operation on D O(k)

D := DDk set of k-ary operations on D

OD :=

k∈N+ O(k) D

set of all finitary operations on D Finitary relations For m ∈ N+ subsets ̺ ⊆ Dm are m-ary relations on D R(m)

D

:= P (Dm) set of m-ary relations on D RD :=

m∈N+ R(m) D

set of all finitary relations on D

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

x0,0 · · · x0,n−1 . . . ... . . . xm−1,0 · · · xm−1,n−1

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

x0,0 · · · x0,n−1 . . . ... . . . xm−1,0 · · · xm−1,n−1 ∈ ̺ · · · ∈ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

f ( x0,0 · · · x0,n−1 ) . . . ... . . . f ( xm−1,0 · · · xm−1,n−1 ) ∈ ̺ · · · ∈ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

f ( x0,0 · · · x0,n−1 ) = . . . ... . . . f ( xm−1,0 · · · xm−1,n−1 ) = ∈ ̺ · · · ∈ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

f ( x0,0 · · · x0,n−1 ) = y0 . . . ... . . . . . . f ( xm−1,0 · · · xm−1,n−1 ) = ym−1 ∈ ̺ · · · ∈ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

f ( x0,0 · · · x0,n−1 ) = y0 . . . ... . . . . . . f ( xm−1,0 · · · xm−1,n−1 ) = ym−1 ∈ ̺ · · · ∈ ̺ ∈ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Preservation condition

m, n ∈ N+, f ∈ O(n)

D , ̺ ∈ R(m) D

f ( x0,0 · · · x0,n−1 ) = y0 . . . ... . . . . . . f ( xm−1,0 · · · xm−1,n−1 ) = ym−1 ∈ ̺ · · · ∈ ̺ ∈ ̺ Truth of this condition: f ⊲ ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Polymorphisms and invariant relations

For F ⊆ OD and Q ⊆ RD: InvD F := {̺ ∈ RD | ∀f ∈ F : f ⊲ ̺} PolD Q := {f ∈ OD | ∀̺ ∈ Q : f ⊲ ̺} closure operators F → PolD InvD F Q → InvD PolD Q

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Connection to clones

Lemma Q ⊆ RD = ⇒ PolD Q is a clone.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Connection to clones

Lemma Q ⊆ RD = ⇒ PolD Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ OD a clone = ⇒ F = PolD Q for Q = InvD F.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Connection to clones

Lemma Q ⊆ RD = ⇒ PolD Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ OD a clone = ⇒ F = PolD Q for Q = InvD F. Consequence Every clone can be described by relations.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Connection to clones

Lemma Q ⊆ RD = ⇒ PolD Q is a clone. Theorem (Bodnarčuk, Kalužnin, Kotov, Romov 69, Geiger 68) D finite, F ⊆ OD a clone = ⇒ F = PolD Q for Q = InvD F. Consequence Every clone can be described by relations. Idea Reduction of complexity by confining the allowed relations

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Outline

1

Galois theory for clones

2

Clausal relations and clausal clones

3

Maximal clones/C-clones

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Clausal relations

From now on D = {0, . . . , n − 1} finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Clausal relations

From now on D = {0, . . . , n − 1} finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1. Definition (Clausal relation) p, q ∈ N+, a = (a1, . . . , ap) ∈ Dp, b = (b1, . . . , bq) ∈ Dq. Ra

b :=

• (x, y) ∈ Dp+q
• p
• i=1

xi ≥ ai ∨

q

• j=1

yj ≤ bj

• .

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Clausal relations

From now on D = {0, . . . , n − 1} finite! Chain 0 ≤ 1 ≤ 2 ≤ · · · ≤ n − 1. Definition (Clausal relation) p, q ∈ N+, a = (a1, . . . , ap) ∈ Dp, b = (b1, . . . , bq) ∈ Dq. Ra

b :=

• (x, y) ∈ Dp+q
• p
• i=1

xi ≥ ai ∨

q

• j=1

yj ≤ bj

• .

Special case: binary clausal relation p = q = 1, a = (a) ∈ D1, b = (b) ∈ D1. R(a)

(b) =

• (x, y) ∈ D2

x ≥ a ∨ y ≤ b

• .

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

C-clones

Definition (C-clone) = every clone PolD Q, where Q is a set of clausal relations.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

C-clones

Definition (C-clone) = every clone PolD Q, where Q is a set of clausal relations. C-clones form a lattice w.r.t. ⊆.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, 1}

OD M = PolD ≤ c0, c1, ∨OD c0, c1, ∧OD CD

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, 1}

OD M = PolD ≤ c0, c1, ∨OD c0, c1, ∧OD CD maximal clone in LD

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, . . . , n − 1}, n ≥ 3

contains countably infinite descending chains

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, . . . , n − 1}, n ≥ 3

contains countably infinite descending chains OD PolD R(1)

(1)

PolD R(11)

(11)

PolD R(111)

(111)

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, . . . , n − 1}, n ≥ 3

contains countably infinite descending chains OD PolD R(1)

(1)

PolD R(11)

(11)

PolD R(111)

(111)

no C-clone = a maximal clone [Beh,Var 2014, submitted]

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Lattice of C-clones for D = {0, . . . , n − 1}, n ≥ 3

contains countably infinite descending chains OD PolD R(1)

(1)

PolD R(11)

(11)

PolD R(111)

(111)

no C-clone = a maximal clone [Beh,Var 2014, submitted] = ⇒ every C-clone = OD satisfies F M for some maximal clone M (as OD is finitely generated)

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

The goal

What is the exact relationship? F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal C-clone M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal clone

F ⊆ M

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

The goal

What is the exact relationship? F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal C-clone M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal clone

F ? M

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

The goal

What is the exact relationship? F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal C-clone M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .a maximal clone

F M

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Outline

1

Galois theory for clones

2

Clausal relations and clausal clones

3

Maximal clones/C-clones

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Basics

maximal clones/C-clones are of the form PolD ̺ for some ̺ / ∈ InvD OD.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Basics

maximal clones/C-clones are of the form PolD ̺ for some ̺ / ∈ InvD OD. describing relations are known

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Basics

maximal clones/C-clones are of the form PolD ̺ for some ̺ / ∈ InvD OD. describing relations are known every clone/C-clone F OD is a subset of some maximal clone/C-clone.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Basics

maximal clones/C-clones are of the form PolD ̺ for some ̺ / ∈ InvD OD. describing relations are known every clone/C-clone F OD is a subset of some maximal clone/C-clone. (OD finitely generated / [Var11])

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Maximal C-clones

Theorem (Var11) A C-clone F ⊆ OD is maximal if and only if F = PolD

• R(a)

(b)

• for some a > 0 and some b < n − 1.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

Maximal clones

Theorem (I.G. Rosenberg, 1970) D finite, F ∈ LD maximal iff F = PolD {̺}, for ̺

1 partial order with least and greatest element. 2 graph {(x, f (x)) | x ∈ D} of a prime permutation f . 3 non-trivial equivalence relation on D. 4 affine relation w.r.t. some elementary Abelian p-group on

D, p prime.

5 a central relation of arity h (1 ≤ h < |D|). 6 an h-regular relation (3 ≤ h ≤ |D|). Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ∃ f ∈ PolD R(a)

(b) \ PolD ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ∃ f ∈ PolD R(a)

(b) \ PolD ̺

preferably of small arity

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ∃ f ∈ PolD R(a)

(b) \ PolD ̺

preferably of small arity PolD R(a)

(b) ⊆ PolD ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ∃ f ∈ PolD R(a)

(b) \ PolD ̺

preferably of small arity PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ̺ ∈

• R(a)

(b)

• RD

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

2 Cases

Suppose a > 0, b < n − 1 and ̺ ∈ RD is a Rosenberg relation. PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ∃ f ∈ PolD R(a)

(b) \ PolD ̺

preferably of small arity PolD R(a)

(b) ⊆ PolD ̺

⇐ ⇒ ̺ ∈

• R(a)

(b)

• RD

find a primitive positive formula for ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ = graph(s), s : D − → D

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ = graph(s), s : D − → D f ∈ Pol(n)

D ̺ ⇐

⇒ s (f (x1, . . . , xn)) = f (s(x1), . . . , s(xn))

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ = graph(s), s : D − → D f ∈ Pol(n)

D ̺ ⇐

⇒ s (f (x1, . . . , xn)) = f (s(x1), . . . , s(xn)) ca ∈ Pol(1)

D ̺ ⇐

⇒ s(a) = a.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ = graph(s), s : D − → D f ∈ Pol(n)

D ̺ ⇐

⇒ s (f (x1, . . . , xn)) = f (s(x1), . . . , s(xn)) ca ∈ Pol(1)

D ̺ ⇐

⇒ s(a) = a. prime permutations have no fixed points

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ = graph(s), s : D − → D f ∈ Pol(n)

D ̺ ⇐

⇒ s (f (x1, . . . , xn)) = f (s(x1), . . . , s(xn)) ca ∈ Pol(1)

D ̺ ⇐

⇒ s(a) = a. prime permutations have no fixed points = ⇒ ca ∈ PolD R(a)

(b) \ PolD ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ ⊆ Dm totally reflexive, m ≥ 3

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ ⊆ Dm totally reflexive, m ≥ 3 ∨D ∈ PolD R(a)

(b) \ PolD ̺

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ ⊆ Dm totally reflexive, m ≥ 3 ∨D ∈ PolD R(a)

(b) \ PolD ̺

h-regular rel’s are totally reflexive, h-ary, h ≥ 3

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s central rel’s equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ ⊆ Dm totally reflexive, m ≥ 3 ∨D ∈ PolD R(a)

(b) \ PolD ̺

h-regular rel’s are totally reflexive, h-ary, h ≥ 3 central rel’s are totally reflexive by definition

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ affine rel w.r.t. GF(p)-vector space structure on D

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ affine rel w.r.t. GF(p)-vector space structure on D f ∈ Pol(n)

D ̺ ⇐

⇒ f : Dn − → D affine linear function

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ affine rel w.r.t. GF(p)-vector space structure on D f ∈ Pol(n)

D ̺ ⇐

⇒ f : Dn − → D affine linear function = ⇒ Ker (f − f (0, . . . , 0)) = f −1 [{f (0, . . . , 0)}] ≤ Dn

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ affine rel w.r.t. GF(p)-vector space structure on D f ∈ Pol(n)

D ̺ ⇐

⇒ f : Dn − → D affine linear function = ⇒ Ker (f − f (0, . . . , 0)) = f −1 [{f (0, . . . , 0)}] ∼ = (GF(p))t

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ affine rel w.r.t. GF(p)-vector space structure on D f ∈ Pol(n)

D ̺ ⇐

⇒ f : Dn − → D affine linear function = ⇒ Ker (f − f (0, . . . , 0)) = f −1 [{f (0, . . . , 0)}] ∼ = (GF(p))t ∃f ∈ Pol(1)

D R(a) (b) : |f −1 [{f (0, . . . , 0)}]| is no power of p.

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ bounded partial order relation

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ bounded partial order relation technical case distinction showing

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆ ̺ bounded partial order relation technical case distinction showing ∃f ∈ Pol(≤2)

D

R(a)

(b) \ PolD ̺ (i.e. f not monotone w.r.t. ̺)

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

∃f / ∈ PolD ̺

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Galois theory C-clones Max clones

OD bounded

• rders

∃f / ∈ PolD ̺

affine rel’s

∃f / ∈ PolD ̺

graphs of prime perm

ca / ∈ PolD ̺

h-regular rel’s

∨D / ∈ PolD ̺

central rel’s

ar (̺) ≤ 2

equiva- lences next PolD

• R(a)

(b)

? ⊆

Mike Behrisch Relationship of max clones / C-clones

Theorem

Binary central relations Unary central rel’s ∅ ̺ D Equivalence relations θ

Theorem

Binary central relations Unary central rel’s ∅ ̺ D Equivalence relations θ ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D Equivalence relations θ ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D Equivalence relations θ ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2 ∅ ̺ D central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b > 1 and ̺ = {0, . . . , b} ∪ {a, . . . , n − 1}

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D a − b > 1, ̺ = ↓ b ∪ ↑ a Equivalence relations θ ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2 ∅ ̺ D central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b > 1 and ̺ = {0, . . . , b} ∪ {a, . . . , n − 1}

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D a − b > 1, ̺ = ↓ b ∪ ↑ a Equivalence relations θ ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2 ∅ ̺ D central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b > 1 and ̺ = {0, . . . , b} ∪ {a, . . . , n − 1} ∆ θ D2 equivalence, then PolD R(a)

(b) ⊆ PolD θ ⇐

⇒ a − b = 1 and D/θ = {{0, . . . , b} , {a, . . . , n − 1}}

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D a − b > 1, ̺ = ↓ b ∪ ↑ a Equivalence relations θ a − b = 1, D/θ = {↓ b, ↑ a} ∆ ̺ D2 central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b < 1 and ̺ = {0, . . . , b}2 ∪ {a, . . . , n − 1}2 ∅ ̺ D central, then PolD R(a)

(b) ⊆ PolD ̺ ⇐

⇒ a − b > 1 and ̺ = {0, . . . , b} ∪ {a, . . . , n − 1} ∆ θ D2 equivalence, then PolD R(a)

(b) ⊆ PolD θ ⇐

⇒ a − b = 1 and D/θ = {{0, . . . , b} , {a, . . . , n − 1}}

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D a − b > 1, ̺ = ↓ b ∪ ↑ a Equivalence relations θ a − b = 1, D/θ = {↓ b, ↑ a} Theorem For finite D every maximal C-clone is contained in exactly one maximal clone.

Theorem

Binary central relations a − b < 1, ̺ = (↓ b)2 ∪ (↑ a)2 Unary central rel’s ∅ ̺ D a − b > 1, ̺ = ↓ b ∪ ↑ a Equivalence relations θ a − b = 1, D/θ = {↓ b, ↑ a} Theorem For finite D every maximal C-clone is contained in exactly one maximal clone. Details, D = {0, . . . , n − 1} n = 2 PolD R1

0 = M = PolD ≤2.

n ≥ 3 a − b < 1 = ⇒ PolD R(a)

(b) ⊆ PolD

• (↓ b)2 ∪ (↑ a)2

a − b > 1 = ⇒ PolD R(a)

(b) ⊆ PolD {↓ b ∪ ↑ a}

a − b = 1 = ⇒ PolD R(a)

(b) ⊆ PolD {θ} , D/θ = {↓ b, ↑ a}

Galois theory C-clones Max clones

Final clause

Thank ∧ (¬me) ∧ for ∧ your ∧ (¬inattention) .

Mike Behrisch Relationship of max clones / C-clones

Completeness criterion

Corollary Let D = {0, . . . , n − 1}, n ≥ 3, F be a clausal clone. If ∀ 0 ≤ b < n − 1 ∃ f ∈ F : f ⊲ θb, D/θb = [0, . . . , b|b + 1, . . . , n − 1], and ∀ 0 < a ≤ b < n − 1 ∃ f ∈ F : f ⊲ (↓ b)2 ∪ (↑ a)2, and ∀ 0 ≤ b ≤ n − 3 ∀ 2 ≤ k ≤ n − 1 − b ∃ f ∈ F : f ⊲ ↓ b ∪ ↑ (b + k); then F = OD.

Mike Behrisch Relationship of max clones / C-clones

References

Mike Behrisch and Edith Mireya Vargas, C-clones and C-automorphism groups, Contributions to general algebra 19, Heyn, Klagenfurt, 2010, Proceedings of the Olomouc Workshop 2010 on General Algebra., pp. 1–12. MR 2757766 Mike Behrisch and Edith Vargas-García, On the relationship of maximal C-clones and maximal clones, Preprint MATH-AL-01-2014, TU Dresden, Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germany, January 2014, online available at http://nbn-resolving.de/urn:nbn:de:bsz: 14-qucosa-131431. Edith Vargas, Clausal relations and C-clones, Discuss.

• Math. Gen. Algebra Appl. 30 (2010), no. 2, 147–171,

10.7151/dmgaa.1167. Edith Mireya Vargas, Clausal relations and C-clones, Dissertation, TU Dresden, 2011, online available at http://nbn-resolving.de/urn:nbn:de:bsz: 14-qucosa-70905.

Mike Behrisch Relationship of max clones / C-clones