SZTE Bolyai Intézet · 2012-06-25 · BackgroundSelf-Dual unctionsFMinimal ClonesurtherF ProspectiveMain Ideas Lattice of Clones. De nitions Ek = f0;1;2;:::;k 1g: Pn k = ffjf : En

Background Self-Dual Functions Minimal Clones Further Prospective Main Ideas

On the Lattice of Clones on Three Elements

Dmitriy Zhuk

[email protected]

Department of Mathematics and Mechanics

Moscow State University

Conference on Universal Algebra and Lattice Theory

Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Outline

1 Background

2 Self-Dual Functions

3 Minimal Clones

4 Further Prospective

5 Main Ideas



Lattice of Clones.

De�nitions

Ek = {0,1,2, . . . , k − 1}.Pn

k = {f | f : Enk → Ek}, Pk =

⋃n≥1

Pnk .

Pk is the set of all functions in k -valued logic.

F ⊆ Pk is a clone if F is closed under superposition and

contains all projections.

Jk is the set of all projections.

The clones form an algebraic lattice whose least element is Jkand whose greatest element is Pk .



Background

All clones in two-valued logic were described (E. Post, 1921,

1941)

There exists a continuum of clones in k -valued logic for

k ≥ 3 (Ju. I. Janov, A.A. Muchnik, 1959).

All maximal (also known as precomplete) clones in

three-valued logic were found (S.V. Jablonskij, 1955)

All minimal clones in three-valued logic were found

(B.Cs�ak�any, 1983)



Background

All 158 submaximal clones in three-valued logic were

described (D. Lau, H. Machida, J. Demetrovics, L. Hannak,

S. S. Marchenkov, J. Bagyinszki).

All maximal clones except the clone of all linear functions

contain a continuum of subclones (J. Demetrovics,

L. Hannak, S. S. Marchenkov, 1983).

The cardinality of the set of all clones in a given

submaximal clone in three-valued logic was found

(A. Bulatov, A. Krokhin, K. Sa�n, E. Sukhanov).



Background

All 158 submaximal clones in three-valued logic were

described (D. Lau, H. Machida, J. Demetrovics, L. Hannak,

S. S. Marchenkov, J. Bagyinszki).

All maximal clones except the clone of all linear functions

contain a continuum of subclones (J. Demetrovics,

L. Hannak, S. S. Marchenkov, 1983).

The cardinality of the set of all clones in a given

submaximal clone in three-valued logic was found

(A. Bulatov, A. Krokhin, K. Sa�n, E. Sukhanov).



Self-Dual Functions

De�nitions

An n-ary function in three-valued logic is called self-dual if

f (x1 + 1, x2 + 1, . . . , xn + 1) = f (x1, x2, . . . , xn) + 1

for all x1, x2, . . . , xn ∈ {0,1,2}.+ is addition modulo 3.

Self-dual functions are functions that preserve the relation(0 1 21 2 0

).

The set of all self-dual functions is a maximal clone in

three-valued logic



Main Result

Many clones of self-dual functions were found

(S. S. Marchenkov, J. Demetrovich, L. Hannak, 1980).

There exists a continuum of clones of self-dual functions

(S. S. Marchenkov, 1983)

Main Result

All clones of self-dual functions in three-valued logic are

described

This is the �rst maximal clone besides the clone of all linear

functions that has such description.



Main Result

Many clones of self-dual functions were found

(S. S. Marchenkov, J. Demetrovich, L. Hannak, 1980).

There exists a continuum of clones of self-dual functions

(S. S. Marchenkov, 1983)

Main Result

All clones of self-dual functions in three-valued logic are

described

This is the �rst maximal clone besides the clone of all linear

functions that has such description.





Meaning of Lines

solid line

M

N M is a maximal clone in N.

dotted line

M

N M ⊂ N, sublattice between M and N is a

countable chain.

dotted line inside a dotted ellipse

M

N M ⊂ N, sublattice between M and N is not a

chain.





Predicates

De�nition

A mapping Enk → {0,1} is called an n-ary predicate.

Rnk = {ρ| ρ : En

k → {0,1}},Rk =⋃

n≥1Rn

k .

De�nition

Inv(M) is the set of all predicates that are preserved by

functions from M.

De�nition

Pol(S) is the set of all functions that preserve predicates from S.



Duality

Let σ : E3 −→ E3, σ(0) = 1, σ(1) = 0, σ(2) = 2.

De�nition

By f ∗ we denote the function that is dual (with respect to σ) tof :

f ∗(x1, . . . , xn) := σ−1(f (σ(x1), σ(x2), . . . , σ(xn))).

Suppose M ⊆ P3, then put M∗ := {f ∗ | f ∈ M}.

Clones on the left side of the �gure are dual (with respect

to σ) to the clones on the right side of the �gure





Set of Predicates Π

Suppose

m ∈ {1,2,3, . . . , }, n ∈ {0,1,2,3, . . . , }.A1,A2, . . . ,Am ⊆ {1,2, . . . ,n},A1 ∪ A2 ∪ . . . ∪ Am = {1,2, . . . ,n}.

In case of n = 0 we have A1 = A2 = . . . = Am = ∅.Let πA1,A2,...,Am be the predicate of arity m + n such that

πA1,A2,...,Am (a1, . . . ,am,b1, . . . ,bn) = 1

i� the following conditions hold:

1 ∀i(ai = 1 ∨ (ai = 0 ∧ (∀j ∈ Ai : bj ∈ {0,1})));

2 at least one of the values a1, . . . ,am,b1, . . . ,bn is not equal

to 0.

The set of all such predicates we denote by ΠDmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Quasiorder

We de�ne a quasiorder . on the set Π;

De�nition

F ⊆ Π is a down-set if the following condition holds

∀ρ ∈ F ∀σ ∈ Π (σ . ρ =⇒ σ ∈ F ).



Continual Class of Clones

For F ⊆ Π we put

Clone(F ) = Pol(

F ∪{(

0 1 21 2 0

),

(0 0 1 1 10 1 0 1 2

)}),

Clone∗(F ) = (Clone(F ))∗,

The continual class of clones consists of all clones

Clone(F ), Clone∗(F ),

where F ⊆ Π is a nonempty down-set.



Continual Class of Clones

Theorem

Suppose F1,F2 are nonempty down-sets, then

Clone(F1) ⊆ Clone(F2)⇐⇒ F1 ⊇ F2.

Corollary

Suppose F1,F2 are nonempty down-sets, then

F1 6= F2 =⇒ Clone(F1) 6= Clone(F2).

Each clone from continual class is uniquely de�ned by a

nonempty down-set F ⊆ Π.











Pairwise Inclusion of the Clones Into Each Other

For the �nite and countable classes of clones pairwise

inclusion is shown by the graph.

For the continual class of clones we formulate theorems that

describe pairwise inclusion.



Bases of Clones

De�nition

B is a basis of M i� [B] = M and ∀B′ ⊂ B : [B′] 6= M.

It is proved that every clone of self-dual functions has a

basis (�nite or in�nite)

All clones with a �nite basis (or �nitely generated clones)

are found

All clones from �nite and countable classes have a �nite

basis (are �nitely generated)

A basis for each clone from the �nite and countable classes

is found

For the continual class of clones we formulate a simple

criterion for the existence of a �nite basis



Relation Degree

De�nition

Relation degree of a clone is the minimal arity of relations such

that the clone can be de�ned as the set of all functions that

preserve these relations.

The relation degree of each clone of self-dual functions is

found

The values of relation degree prove that our description is

minimal. That is, these clones can not be de�ned by

relations of smaller arity.



Cardinality of principal �lters and principal ideals

De�nition

L3 is the set of all clones of self-dual functions,

L↑3(M) := {M ′ ∈ L3|M ⊆ M ′),

L↓3(M) := {M ′ ∈ L3|M ′ ⊆ M).

For each clone F ∈ L3 the cardinality of L↑3(F ) is found.

For each clone F ∈ L3 the cardinality of L↓3(F ) is found.

L↓3(M) is continuum for every M = Clone(F ) such that

M 6= Clone(Π). L↓3(Clone(Π)) = 4.





Minimal Clones

De�nition

A clone A 6= Jk is called minimal, if the following condition

holds:

∀f ∈ A \ Jk : [{f} ∪ Jk ] = A.

Examples of minimal clones in P3

[{max(x , y)}], [{2x + 2y}], [{0, x}].

I. Rosenberg classi�ed all minimal clones in Pk (1983).

B.Cs�ak�any found all minimal clones in P3 (1983).

All minimal clones in P4 were found (P�oshel, Kaluzhnin,

Szczepara, Waldhauser, Jezek, Quackenbush, Karsten

Sch�olzer, 2012).



Rosenberg Classi�cation of Minimal Clones

Theorem

Each minimal clone in Pk can be presented as [Jk ∪ {f}] where1 f is an unary function.

2 f is a binary idempotent function

f (a,a) = a for every a ∈ E3.

3 f is a majority function

∀a,b ∈ E3 : f (a,a,b) = f (a,b,a) = f (b,a,a) = a4 f is a minority function

∀a,b ∈ E3 : f (a,a,b) = f (a,b,a) = f (b,a,a) = b5 f is a semiprojection to the i-th variable

f ∈ Pnk , 3 ≤ n ≤ k , for every tuple (a1,a2, . . . ,an) satisfying

|{a1,a2, . . . ,an}| < n we have f (a1,a2, . . . ,an) = ai .

We will assume that i = 1 for all occurring semiprojections.Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Minimal Clones on Three elements (found by B.Cs�ak�any)

P3 has exactly 84 minimal clones. One obtains every one of these clones byapplying inner automorphisms of P3 to exactly one of the following clones:

a) Minimal clones generated by unary functionsx c0 c1 c2 c30 1 1 0 11 1 0 1 22 1 2 1 0

b) Minimal clones generated by idempotent binary functionsx y b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b120 1 1 0 1 1 1 1 0 0 0 0 1 21 0 1 1 1 1 1 1 1 1 1 1 1 20 2 1 0 0 0 0 0 0 1 0 1 0 12 0 1 1 0 2 0 2 0 1 1 2 0 11 2 1 1 1 1 1 1 1 1 1 0 2 02 1 1 1 1 1 2 2 1 2 2 2 2 0

c) Minimal clones generated by majority functions m1,m2,m3 andsemiprojections s1, s2, s3, s4, s5

x y z m1 m2 m3 s1 s2 s3 s4 s50 1 2 1 0 0 1 0 0 1 10 2 1 1 1 0 1 0 0 1 21 0 2 1 1 1 1 1 1 0 01 2 0 1 0 1 1 1 1 0 22 0 1 1 0 2 1 1 0 2 02 1 0 1 1 2 1 1 1 2 1



Main Problem

How many clones contain a given minimal clone on three

elements?

For minimal clones generated by majority functions the resultfollows from Baker-Pixley Theorem (K.A. Baker, A. F. Pixley,1975)

Clones containing s5 and clones containing b12 are �nitelygenerated (S. S. Marchenkov, 1984)

The set of all clones containing b12 is �nite and completelydescribed (I. Rosenberg, 1995).

For the minimal clones generated by unary functions thisproblem was solved by J. Pantovi�c and D. Voivodi�c (2000).

The set of all clones containing a function f has continuumcardinality for every f ∈ {s1,s2,b1,b2,b4,b6,b8,b9}. The set ofall clones containing a function g is at least countable for everyg ∈ {s3,s4,b3,b5,b7,b10} (J. Pantovi�c, D. Voivodi�c, 2000).



Main Problem


elements?








Main Problem


elements?








Main Problem


elements?








Main Problem


elements?








Main Problem


elements?








Main Result

Theorem

Suppose M = [J3 ∪ {f}] is a minimal clone in P3. The set of allclones containing M

is �nite if f ∈ {c3,b12,m1,m2,m3},is countable if f ∈ {s4,s5},has continuum cardinality if

f ∈ {c0,c1,c2,b1,b2,b3, . . . ,b10,b11,s1,s2,s3}.



Finite and countable cases

Finite cases (except clones generated by majority functions)

c3(x) = x + 1;

b12(x , y) = 2x + 2y .

Countable cases

s4(x , y , z) =

{2x + 1, if |{x , y , z}| = 3;x , otherwise.

;

s5(x , y , z) =

{y , if |{x , y , z}| = 3;x , otherwise.

.



The Lattice of Clones containing x + 1



The Lattice of Clones containing 2x + 2y



Clones containing a majority function

P3

How many clones

on three elements

contain a majority function?

1 918 040




P3

How many clones

on three elements


1 918 040




P3

How many clones

on three elements


1 918 040




P3

How many clones

on three elements


1 918 040




P3

How many clones

on three elements


1 918 040




P3

How many clones

on three elements


1 918 040Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Clones containing the semiprojection s4 or s5

For m,n ∈ {1,2,3, . . .} letρm,n = ({0,1}m × {0,2}n) \ {(0,0,0, . . . ,0)}

Let S = {ρm,n|m,n ∈ {1,2,3, . . .}}.

Lemma

Every semiprojection in P3 preserves the relation ρm,n.

Lemma

Suppose 1 ≤ m′ ≤ m, 1 ≤ n′ ≤ n, then ρm′,n′ ∈ [{ρm,n}].

De�nition

A set H ⊆ S is called closed if

ρm,n ∈ H ⇒ (∀m′ ≤ m ∀n′ ≤ n ρm′,n′ ∈ H).

Lemma

Suppose H1,H2 ⊆ S are closed, then

H1 6= H2 ⇒ Pol(H1) = Pol(H2).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1

2

3

4

5

6

7

8

9

10

11

12

13

14

n

m

ρ15,8

ρ4,5

ρ11,4

ρm,n

ρ11,8

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ8,7

ρ8,8

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ9,7

ρ9,8

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ10,7

ρ10,8

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ11,7

ρ11,8

ρ3,12

ρ7,11

ρ17,6

ρ8,5

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ1,12

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ2,12

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ3,12

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ1,7

ρ1,8

ρ1,9

ρ1,10

ρ1,11

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ2,7

ρ2,8

ρ2,9

ρ2,10

ρ2,11

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ3,7

ρ3,8

ρ3,9

ρ3,10

ρ3,11

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ4,7

ρ4,8

ρ4,9

ρ4,10

ρ4,11

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ5,7

ρ5,8

ρ5,9

ρ5,10

ρ5,11

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ6,7

ρ6,8

ρ6,9

ρ6,10

ρ6,11

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ7,7

ρ7,8

ρ7,9

ρ7,10

ρ7,11

ρ1,1

ρ1,2

ρ1,3

ρ1,4

ρ1,5

ρ1,6

ρ2,1

ρ2,2

ρ2,3

ρ2,4

ρ2,5

ρ2,6

ρ3,1

ρ3,2

ρ3,3

ρ3,4

ρ3,5

ρ3,6

ρ4,1

ρ4,2

ρ4,3

ρ4,4

ρ4,5

ρ4,6

ρ5,1

ρ5,2

ρ5,3

ρ5,4

ρ5,5

ρ5,6

ρ6,1

ρ6,2

ρ6,3

ρ6,4

ρ6,5

ρ6,6

ρ7,1

ρ7,2

ρ7,3

ρ7,4

ρ7,5

ρ7,6

ρ8,1

ρ8,2

ρ8,3

ρ8,4

ρ8,5

ρ8,6

ρ9,1

ρ9,2

ρ9,3

ρ9,4

ρ9,5

ρ9,6

ρ10,1

ρ10,2

ρ10,3

ρ10,4

ρ10,5

ρ10,6

ρ11,1

ρ11,2

ρ11,3

ρ11,4

ρ11,5

ρ11,6

ρ12,1

ρ12,2

ρ12,3

ρ12,4

ρ12,5

ρ12,6

ρ13,1

ρ13,2

ρ13,3

ρ13,4

ρ13,5

ρ13,6

ρ14,1

ρ14,2

ρ14,3

ρ14,4

ρ14,5

ρ14,6

ρ15,1

ρ15,2

ρ15,3

ρ15,4

ρ15,5

ρ15,6

ρ16,1

ρ16,2

ρ16,3

ρ16,4

ρ16,5

ρ16,6

ρ17,1

ρ17,2

ρ17,3

ρ17,4

ρ17,5

ρ17,6

S11

S11

S10

S10

S9

S9

S8

S8

S7

S7S6

S6

S5

S5

S4

S4

S3

S3

S2

S2

S1

S1





Properties of clones containing s4 or s5.

Theorem (S. S.Marchenkov, 1984)

Every clone in P3 containing s5 is �nitely generated.

Theorem

Every clone in P3 containing s4 is �nitely generated.



Open Problems

Problem 1

Find all minimal clones on k ≥ 4 elements such that the set of

clones containing this minimal clone is �nite or countable.

Problem 2

Does there exist a minimal clone in k -valued logic that is not

�nitely related (cannot be de�ned by a relation)?

A clone M is called essentially minimal, if every essential

function in M generates M.

Problem 3

Find all essentially minimal clones on k ≥ 3 elements such that

the set of clones containing this minimal clone is �nite or

countable.



Clones containing a majority operation

P3

How many clones

on three elements


1 918 040Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Maximal clones on three elements

The number of clones containing a majority operation

Pol(0) Pol({0,1}) Pol(

0 1 21 2 0

)Pol(

0 1 2 0 10 1 2 1 0

)1 744 466 1 722 818 17 36 942

Pol(

0 1 2 0 0 10 1 2 1 2 2

)Pol(

0 1 2 0 0 1 20 1 2 1 2 0 0

)1 546 84 201



Minimal clones on three elements

The number of clones containing a majority operation

a) Minimal clones generated by unary functionsc0 c1 c2 c3

52 530 83 67 837 2

b) Minimal clones generated by idempotent binary functionsb1 b2 b3 b4 b5 b6

144 099 103 572 375 270 371 876 107 444 425 584b7 b8 b9 b10 b11 b12

312 915 38 981 82 786 996 77 457 17

c) Minimal clones generated by majority functionsm1 m2 m3

928 767 325 581 99 152

d) Minimal clones generated by semiprojectionss1 s2 s3 s4 s5

217 601 469 196 196 315 55 898 14 832Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Decidability Problems

Problem 1

Given a �nite set of relations S; decide whether Pol(S) is�nitely generated.

Problem 2

Given a �nite set of functions M; decide whether [M] is �nitelyrelated.

Problem 3

Given a �nite set of functions M; a �nite set of relations S;decide whether [M] = Pol(S).



Partial Results

Problems 1, 2, and 3 are decidable for

clones in P2

clones of linear functions in P3

clones of self-dual functions in P3

clones containing a majority function

clones containing 2x + 2y or x + 1 in three-valued logic.

clones containing s4 or s5 in three-valued logic.

clones containing b11 in three-valued logic.



Main ideas

We essentially use a Galois connection between clones and

relational clones. We operate with relations instead of

operating with functions.

We consider only predicates that cannot be presented as a

conjunction of predicates with smaller arities. These

predicates are called essential.

We investigate the lattice not from the top but from the

bottom; select a clone M near the bottom of the lattice (for

example a minimal clone), then describe the set of all

essential predicates from Inv(M).

We de�ne a closure operator on the set of all essential

predicates. Using this closure operator we describe closed

sets of predicates.



Essential predicates

De�nition

A predicate ρ is called essential i� ρ cannot be presented as a

conjunction of predicates with arity less then the arity of ρ. ByR̃k we denote the set of all essential predicates

De�nition

A tuple (a1, . . . ,an) is called essential for ρ if there existsb1, . . . ,bn such that

ρ(a1, . . . ,an) = 0,∀i ρ(a1, . . . ,ai−1,bi ,ai+1, . . . ,an) = 1.



Examples

Essential Predicates

ρ(x , y , z) = 1⇐⇒ (x ⊕ y ⊕ z = 1)(0,0,0) is an essential tuple(

0 0 10 1 1

)(1,0) is an essential tuple(

0 10 1

)(0,1) is an essential tuple

Not Essential Predicates

ρ(x , y , z) = (x ≤ y) ∧ (y ≤ z).



Properties of essential predicates

Theorem

The following conditions are equivalent:

ρ is an essential predicate;

there exists an essential tuple for ρ.

Theorem

[S ∩ R̃k ] = S for every S = [S] ⊆ Rk .

Every closed set S ⊆ Rk can be described by the set S ∩ R̃kof all essential predicates of S.



Example of Usage

Lemma

Suppose m(x , y , z) is a majority function, m ∈ Pol(ρ), ρ is anessential predicate. Then arity(ρ) ≤ 2.

Proof

Assume that (a1,a2, . . . ,an) is an essential tuple for ρ, n > 2.

m

b1 a1 a1a2 b2 a2a3 a3 b3a4 a4 a4. . . . . . . . .an an an

=

a1a2a3a4. . .an

∈ ρ

This contradicts the condition ρ(a1, . . . ,an) = 0.



Majority Function in P2.



Function x ∨ yz.

Let

ρ∨,n(x1, x2, . . . , xn) = 1⇐⇒ (x1 = 1) ∨ (x2 = 1) ∨ . . . ∨ (xn = 1).

C =

{(0),(1),

(0 10 1

),

(0 0 10 1 1

),

(0 1 10 0 1

), ρ∨,2, ρ∨,3, ρ∨,4, . . .

}.

Lemma

Inv(x ∨ yz) ∩ R̃2 = C.

All clones containing x ∨ yz can be described by predicates from

the set C.



Function x ∨ yz.

Proof

Suppose ρ ∈ Inv(x ∨ yz) ∩ R̃2, the arity of ρ is equal to n, wheren ≥ 3. Let us show that an essential tuple for ρ cannot contain1. Assume that (1,0,0, . . . ,0) is an essential tuple for ρ. Then

0000. . .0

∨

1100. . .0

∧

1010. . .0

=

1000. . .0

∈ ρThis contradicts the de�nition of an essential tuple. Hence

(0,0,0, . . . ,0) is the only essential tuple for ρ. Then, obviouslyρ is equal to ρ∨,n.



C1 = P2, C3 = Pol(0), A1 = Pol

(0 0 10 1 1

)A3 = Pol

({(0 0 10 1 1

),(0)})

;

C2 = Pol(1), C4 = Pol

({(1),(0)})

,

A2 = Pol({(

1),

(0 0 10 1 1

)}),

A3 = Pol({(

1),

(0 0 10 1 1

),(0)})

;

F n4 = Pol (ρ∨,n) , F n

1 = Pol({ρ∨,n,

(0)})

,

F n3 = Pol

({ρ∨,n,

(0 0 10 1 1

)}),

F n2 = Pol

({ρ∨,n,

(0 0 10 1 1

),(0)})

;



The set of essential predicates S ⊆ R̃k is called closed if

σ=k , false ∈ S, and the following operations applied to the

predicates from S always give not essential predicate or a

predicate from the set S.1 Permutation of variables.2 Striking of rows (adding existential quanti�ers).3 Identifying of two variables.4 ζ(ρ) = ρ′, whereρ′(x1, x2, . . . , xn−1) = ∃x ρ(x , x , x1, x2, . . . , xn−1).

5 ε(ρ1, ρ2) = ρ, where m ≤ n and

ρ(x1, x2, . . . , xn) = ρ1(x1, x2, . . . , xn) ∧ ρ2(x1, x2, . . . , xm).

6 for 2 ≤ l ≤ k let δl(ρ1, . . . , ρl) = ρ, where

ρ(x1,1, . . . , x1,n1 , . . . , xl,1, . . . , xl,nl ) =

= ∃x(ρ1(x , x1,1, . . . , x1,n1) ∧ . . . ∧ ρl(x , xl,1, . . . , xl,nl )),

(all variables in the right-hand side of the formula are di�erent).Dmitriy Zhuk [email protected] On the Lattice of Clones on Three Elements


Theorem

The set of essential predicates S ⊆ R̃k is closed if and only if

[S] ∩ R̃k = S.

Every closed set of predicates can be uniquely de�ned by

the closed set of essential predicates.

We may give up considering not essential predicates at all.



Examples

Let ρ be a linear order relation on the set Ek . Let us showthat Pol(ρ) is a maximal clone in Pk . Let ρ

′(x , y) = ρ(y , x).Obviously {ρ, ρ′, σ=k , false, true} is closed set of essential

predicates. Then the maximality of Pol(ρ) follows from the

fact that there are no closed subsets that di�er from

{σ=k , false, true}.How to check the equality of the following sets

Pol(

0 1 2 00 1 2 1

),Pol

(0 1 2 0 10 1 2 1 0

)? Using the

closure operator we can show that{(0 1 2 0 10 1 2 1 0

), σ=3 , false, true

}is a closed set. Hence

these sets are di�erent.



Thank you for your attention


Documents

SZTE Bolyai Intézet · 2012-06-25 · BackgroundSelf-Dual unctionsFMinimal ClonesurtherF ProspectiveMain Ideas Lattice of Clones. De nitions Ek = f0;1;2;:::;k 1g: Pn k = ffjf : En