REGULAR ELEMENTS AND GREEN’S RELATIONS IN MENGER ALGEBRAS OF TERMS

REGULAR ELEMENTS AND GREEN’S RELATIONS IN MENGER ALGEBRAS OF TERMS

Klaus Denecke

University of Potsdam, Institute of Mathematics Am Neuen Palais, 14415 Potsdam, Germany

e-mail: kdenecke@rz.uni-potsdam.de

and

Prakit Jampachon

KhonKaen University, Department of Mathematics KhonKaen, 40002 Thailand

e-mail: prajam@.kku.ac.th

Abstract

Defining an (n + 1)-ary superposition operation S

on the set W

(X

) of all n-ary terms of type τ , one obtains an algebra n − clone τ := (W

(X

); S

, x

, . . . , x

) of type (n + 1, 0, . . . , 0). The algebra n − clone τ is free in the variety of all Menger algebras ([9]).

Using the operation S

there are different possibilities to define binary associative operations on the set W

(X

) and on the cartesian power W

(X

)

. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.

Keywords: term, superposition of terms, Menger algebra, regular element, Green’s relations.

2000 Mathematics Subject Classification: 08A35, 08A40, 08A70.

1. Preliminaries

Let τ = (n

)

be a type of algebras with an n

-ary operation symbol f

for every i in some index set I. For each n ≥ 1 let X

= {x

, . . . x

} be an n-element alphabet. We denote by W

(X

) the set of all n-ary terms of type τ . It is very common to illustrate terms by tree diagrams. Consider for example the type τ = (2) with a binary operation symbol f . Then the term

t = f (f (x

, f (x

, x

)), f (x

, f (x

, x

))) corresponds to the tree diagram below.

f

f f

f f

x

x

x

x

x

x

Q Q Q Q

!! !! !!

Q Q Q Q

!! !! !! D D D D D D

  A A   A A

T T

T T T T

  A A   A A

s s

s s

s

s

s

s s s

s

On the set W

(X

) one can define the following (n + 1)-ary superposition operation

S

: W

(X

)

→ W

(X

) by

S

(x

, t

, . . . , t

) := t

, for every 1 ≤ i ≤ n , and

S

(f

(r

, . . . , r

), t

, . . . , t

) := f

(S

(r

, t

, . . . , t

), . . . , S

(r

, t

, . . . , t

)).

Together with the nullary operations x

, . . . , x

one obtains an algebra n − clone τ := (W

(X

); S

, x

, . . . , x

),

which satisfies the identities

(C1)

∼

S

(

∼

Z,

∼

S

(

∼

Y

,

∼

X

, . . . ,

∼

X

), . . . ,

∼

S

(

∼

Y

,

∼

X

, . . . ,

∼

X

))

≈

∼

S

(

∼

S

(

∼

Z,

∼

Y

, . . . ,

∼

Y

),

∼

X

, . . . ,

∼

X

), (C2)

∼

S

(λ

,

∼

X

, . . . ,

∼

X

) ≈

∼

X

for 1 ≤ i ≤ n, (C3)

∼

S

(

∼

X

, λ

, . . . , λ

) ≈

∼

X

for 1 ≤ i ≤ n.

Here

∼

S

is an (n + 1)-ary operation symbol, λ

, . . . , λ

are nullary operation symbols and

∼

Z,

∼

Y

, . . . ,

∼

Y

,

∼

X

, . . . ,

∼

X

are new variables. The algebra n-clone τ is an example of a unitary Menger algebra of rank n. Without the nullary operations one speaks of a Menger algebra of rank n.

Now we consider a type τ

consisting of n-ary operation symbols only.

Let X be an arbitrary countably infinite alphabet of variables and let W

(X) be the set of all terms of type τ . On W

(X) we consider a generalized su- perposition operation S

which is defined for any n ≥ 1, n ∈ ^N

, inductively by the following steps:

Definition 1.1.

(i) If t = x

, 1 ≤ i ≤ n, then S

(x

, t

, . . . , t

) := t

for t

, . . . , t

∈ W

(X).

(ii) If t = x

, n < i, then S

(x

, t

, . . . , t

) := x

. (iii) If t = f

(s

, . . . , s

), then

S

(t, t

, . . . , t

) := f

(S

(s

, t

, . . . , t

), . . . , S

(s

, t

, . . . , t

)).

Then we may consider the algebraic structure

clone

τ

:= W

(X); S

, (x

)

with the universe W

(X), with one (n+1)-ary operation and infinitely many

nullary operations. This algebra is called a Menger algebra with infinitely

many nullary operations. Without the nullary operations we have a Menger

algebra of rank n. It is not difficult to see ([1]) that this algebra satisfies the

axioms

(Cg1) ˜ S

(T, ˜ S

(F

, T

, . . . , T

), . . . , ˜ S

(F

, T

, . . . , T

))

≈ ˜ S

( ˜ S

(T, F

, . . . , F

), T

, . . . , T

).

(Cg2) ˜ S

(T, λ

, . . . , λ

) = T .

(Cg3) ˜ S

(λ

, T

, . . . , T

) = T

for 1 ≤ i ≤ n.

(Cg4) ˜ S

(λ

, T

, . . . , T

) = λ

for j > n.

(Here ˜ S

, λ

are operation symbols corresponding to the operations S

and x

, i ∈ ^N

, respectively and T, T

, F

are new variables.)

In any Menger algebra (G; S

) of rank n a binary operation + can be defined by

x + y := S

(x, y, . . . , y).

It is easy to see that the operation + is associative. The algebra (G; +) is called diagonal semigroup (see e.g., [10]).

On the cartesian power G

one may define a binary operation ∗ by (x

, . . . , x

)∗(y

, . . . , y

) := (S

(x

, y

, . . . , y

), . . . , S

(x

, y

, . . . , y

)). Then (G

; ∗) is also a semigroup.

We notice that (G; +) can be embedded into (G

; ∗). Actually the subsemigroup (4

; ∗|

) of (G

; ∗) where 4

:= {(x, . . . , x) | x ∈ G} is the diagonal of G, is isomorphic to (G; +).

An element x of a semigroup (S; ·) is called regular if there is an element y of the same semigroup such that x · y · x = x. Clearly, every idempotent element is regular. Further, we recall the definition of Green’s relations.

Green’s relations are special equivalence relations which can be defined

on any semigroup or monoid, using the idea of mutual divisibility of ele-

ments. Let S be a semigroup and let S

be the monoid which arises from

S by adding a neutral element. For any semigroup S and any elements a, b

of S, we say aLb if and only if there are c and d in S

such that c · a = b

and d · b = a. Dually, aRb if and only if there are c and d in S

such that

a · c = b and b · d = a. It follows easily from these definitions that L is al-

ways a right congruence, while R is always a left congruence. The relation H

is defined as the intersection of R and L, and the relation D is the join L∨R.

It is easy to see that L ∨ R = R ◦ L = L ◦ R, where ◦ here refers to the usual composition of relations. Finally, the relation J is defined by aJ b if and only if there exist elements c, d, p and q in S

such that a = c · b · d and b = p · a · q. For Green’s relations we will use the following notation, (a, b) ∈ R and aRb and similarly for the other relations. For more information about Green’s relations in general, we refer the reader to [7].

Using Green’s relation R (L) the set of all regular elements of a semi- group S can be described as follows (see e.g., [7]):

Theorem 1.2. Let x be an element of the universe of a semigroup S. Then the following are equivalent:

(i) x is regular.

(ii) [x]

contains an idempotent element.

(iii) [x]

contains an idempotent element.

Then for the set Reg(S) of all regular elements of the semigroup S we have Reg(S) = [ n

[e]

| e ∈ E(S) o

= [ n

[a]

| [a]

∩ E(S) 6= ∅ o .

2. Idempotent elements

In this section we study idempotent terms of type τ with respect to the operations + and ∗. To determine all idempotent elements we need some lemmas. The first lemma answers to the following question:

Let s, t

, . . . t

∈ W

(X

). Under which conditions does there exist an n-ary term q such that s = S

(q, t

, t

, . . . , t

)?

For a term t ∈ W

(X

) we denote by var(t) the set of all variables occurring in t.

Lemma 2.1. Let s, t

, . . . , t

∈ W

(X

). Then s = S

(s, t

, . . . , t

) if and only if for each i, 1 ≤ i ≤ n, if x

∈ var(s), then t

= x

.

P roof. “⇒” Assume that s = S

(s, t

, . . . , t

) and that x

∈ var(s), but

t

6= x

. Then we have to substitute in s for x

a term different from x

and

obtain S

(s, t

, . . . , t

) 6= s, a contradiction.

“⇐” We assume now that from x

∈ var(s) there follows t

= x

. We will show s = S

(s, t

, . . . , t

) by induction on the complexity of the term s. If s = x

∈ X

, then t

= x

and so s = x

= S

(x

, t

, . . . , x

, . . . , t

).

Now assume that for s = f (s

, . . . , s

) for all 1 ≤ i ≤ n we have s

= S

(s

, t

, . . . , t

). Then

S

(s, t

, . . . , t

)

= S

(f (s

, . . . , s

), t

, . . . , t

)

= f (s

(s

, t

, . . . , t

), . . . , S

(s

, t

, . . . , t

))

= f (s

, . . . s

) = s.

Now we solve the equation x

= S

(q, t

, t

, . . . , t

).

Lemma 2.2. Let q, t

, . . . , t

∈ W

(X

). For each i ∈ {1, 2, . . . , n} we have x

= S

(q, t

, . . . , t

) if and only if there is an integer j for 1 ≤ j ≤ n, such that q = x

and t

= x

.

P roof. “⇐” This direction is clear.

“⇒” For the proof of this direction we use the following formula for the operation symbol count op(t) of the term t ([3]). We denote by vb

(s) the number of occurrences of the variable x

in the term s.

op(S

(q, t

, . . . , t

)) =

n

X

k=1

vb

(q)op(t

) + op(q).

(We mention that this formula is also valid for S

and for terms q, t

, . . . , t

∈ W

(X)). Then from 0 = op(x

) = P

k=1

vb

(q)op(t

) + op(q) we obtain op(q) = 0 and q ∈ X

. Let q = x

for some x

∈ X

. Then from x

= S

(q, t

, . . . , t

) = S

(x

, t

, . . . , t

), we have t

= x

.

Now we want to determine all vectors of terms which are idempotent with

respect to the operation ∗. Clearly, an n-vector (q

, . . . , q

) of n-ary terms is

an idempotent element with respect to the operation ∗ if S

(q

, q

, . . . , q

) =

q

for all 1 ≤ i ≤ n.

Then we have:

Theorem 2.3. For n ≥ 1 an n-vector (q

, . . . , q

) is idempotent if and only if the following condition (ID) is satisfied:

(ID) x

∈

n

[

i=1

var(q

) ⇒ q

= x

.

P roof. Assume that (q

, . . . , q

) satisfies condition (ID). Let var(q

) = {x

, . . . , x

} ⊆ X

. By condition (ID) we have q

= x

for all 1 ≤ j ≤ k and then

S

(q

, q

, . . . , q

)

= S

(q

, q

, . . . , q

, q

, q

. . . , q

, q

, q

, . . . , q

)

= S

(q

, q

, . . . , q

, x

, q

. . . , q

, x

, q

, . . . , q

)

= q

by Lemma 2.1. Therefore, (q

, . . . , q

) is idempotent.

Conversely, assume that (q

, . . . , q

) is idempotent. Suppose (q

, . . . , q

) does not satisfy the condition (ID). Then there exist integers i, j, 1 ≤ i, j ≤ n such that x

∈ var(q

), but q

6= x

. Then by Lemma 2.1 we have S

(q

, q

, . . . , q

, . . . , q

) 6= q

which contradicts the idempotency of the vector (q

, . . . , q

). Hence (q

, . . . q

) satisfies the condition (ID).

Considering idempotent elements with respect to the operation +, we obtain:

Corollary 2.4. An element t ∈ W

(X

) is idempotent with respect to + if and only if t = x

for some 1 ≤ i ≤ n.

Now we consider a type τ

consisting of n-ary operation symbols and the binary associative operations ∗

: (W

(X)

)

→ W

(X)

and +

: W

(X)

→ W

(X) defined by

x +

y := S

(x, y, . . . , y) and

(x

, . . . , x

) ∗

(y

, . . . , y

) := (S

(x

, y

, . . . , y

), . . . , S

(x

, y

, . . . , y

)).

Let X

:= X \ X

. Instead of Lemma 2.1 we now have:

Lemma 2.5. Let s, t

, . . . , t

∈ W

(X). If var(s) 6⊆ X

, then s = S

(s, t

, . . . , t

) if and only if for each i with 1 ≤ i ≤ n we have: if x

∈ var(s), then t

= x

.

If var(s) ⊆ X

, then s = S

(s, t

, . . . , t

).

P roof. The proof is clear for var(s) ⊆ X

. If var(s) ∩ X

6= ∅, then we conclude similar as in Lemma 2.1.

It is clear that instead of Lemma 2.2. we have now:

Lemma 2.6. Let q, t

, . . . , t

∈ W

(X). For 1 ≤ i ≤ n we have x

= S

(q, t

, . . . , t

) if and only if there is an integer j with 1 ≤ j ≤ n such that q = x

and t

= x

.

Then we obtain:

Theorem 2.7. An n-vector (q

, . . . , q

) of elements from W

(X) is idem- potent with respect to ∗

if and only if the following condition (ID

) is sat- isfied

(ID)

x

∈

n

[

i=1

var(q

) ∩ X

⇒ q

= x

.

P roof. If (q

, . . . , q

) is idempotent and if x

∈

n

S

i=1

var(q

) ∩ X

, then by Lemma 2.5, q

= x

.

Assume that (ID

) is satisfied. If S

i=1

var(q

)∩X

= ∅, then application of Lemma 2.5 gives S

(q

, q

, . . . , q

) = q

and (q

, . . . , q

) is idempotent. If S

i=1

var(q

) ∩ X

6= ∅ and x

∈ S

i=1

var(q

) for some 1 ≤ j ≤ n, then by (ID

) we obtain q

= x

and Lemma 2.5 gives S

(q

, q

, . . . , q

) = q

and (q

, . . . , q

) is idempotent.

Further we have:

Corollary 2.8. If var(t) ⊆ X

, then the term t ∈ W

(X

) is idempotent

with respect to +

. If var(t) ∩ X

6= ∅, then an element t ∈ W

(X) is

idempotent with respect to +

if and only if t = x

for some 1 ≤ i ≤ n.

3. Regular elements

A vector (q

, . . . , q

) is a regular element with respect to ∗ if there exists a vector (s

, . . . , s

) such that (q

, . . . , q

) = (q

, . . . , q

) ∗ (s

, . . . , s

)∗

(q

, . . . , q

).

By definition of ∗, a vector (q

, . . . , q

) is regular with respect to ∗ if and only if

q

= S

(S

(q

, s

, . . . , s

), q

, . . . , q

) for 1 ≤ i ≤ n.

Moreover we define (see [10]):

Definition 3.1. A vector (q

, . . . , q

) is called a v-regular element if there exists an s such that (q

, . . . , q

) = (q

, . . . , q

) ∗ (s, . . . , s) ∗ (q

, . . . , q

).

By definition of + an element t ∈ W

(X

) is regular with respect to + if there exists a term s ∈ W

(X

) such that t = S

(S

(t, s, . . . , s), t, t, . . . , t).

We define also (see [9]):

Definition 3.2. An element t ∈ W

(X

) is called weakly regular if there exist s

, . . . , s

∈ W

(X

) such that t = S

(S

(t, s

, s

, . . . , s

), t, t, . . . , t).

To determine regular vectors with respect to ∗ we need the following lemma:

Lemma 3.3. Let q

, q

, . . . , q

, s

, s

, . . . , s

∈ W

(X

). Then (q

, . . . , q

) = (q

, . . . , q

) ∗ (s

, . . . , s

) ∗ (q

, . . . , q

) if and only if for all i, j ∈ {1, 2, . . . , n} we have:

x

∈ var(q

) ⇒ ∃ l ∈ {1, 2, . . . , n} (s

= x

and q

= x

).

P roof.

(q

,. . . , q

) ∗ (s

, . . . , s

) ∗ (q

, . . . , q

) = (q

, . . . , q

)

⇔ (S

(q

, S

(s

, q

, . . . , q

), . . . , S

(s

, q

, . . . , q

)), . . . ,

S

(q

, S

(s

, q

, . . . , q

), . . . , S

(s

, q

, . . . , q

))) = (q

, . . . , q

)(using (C1))

⇔ ∀i ∈ {1, 2, . . . , n}(S

(q

, S

(s

, q

, . . . , q

), . . . , S

(s

, q

, . . . , q

)) = q

)

⇔ ∀i ∈ {1, 2, . . . , n}(∀j ∈ {1, 2, . . . , n}

(x

∈ var(q

) ⇒ S

(s

, q

, . . . , q

) = x

)) (by Lemma 2.1)

⇔ ∀i, j ∈ {1, 2, . . . , n}(x

∈ var(q

) ⇒ S

(s

, q

, . . . , q

) = x

)

⇔ ∀i, j ∈ {1, 2, . . . , n}(x

∈ var(q

) ⇒ s

= x

and q

= x

for some l ∈ {1, 2, . . . , n}) (by Lemma 2.2).

Then we obtain the following result:

Theorem 3.4. Let q

, . . . , q

∈ W

(X

). Then (q

, . . . , q

) is a regular vector in (W

(X

)

; ∗) if and only if for any i, j ∈ {1, 2, . . . , n} we have: if x

∈ var(q

) then there exists l ∈ {1, 2, . . . , n} such that q

= x

.

P roof. “⇐” Follows from the previous lemma.

“⇒” Assume that for each i, j ∈ {1, 2, . . . , n} we have: if x

∈ var(q

) then there exists an element l ∈ {1, 2, . . . , n} such that q

= x

. For each 1 ≤ j ≤ n we obtain: if x

∈

n

S

i=1

var(q

), then there exists an element l with 1 ≤ l ≤ n such that q

= x

. We select an index l

, so that q

= x

. Since

n

S

i=1

var(q

) 6= ∅, let x

∈

n

S

i=1

var(q

) be fixed. Then we define s

for 1 ≤ j ≤ n as follows:

s

=



 



 



x

if x

∈

n

[

i=1

var(q

)

x

otherwise.

It is not difficult to see that for each 1 ≤ i, j ≤ n we get: if x

∈ var(q

), then there exists an element 1 ≤ l ≤ n such that s

= x

and q

= x

. By Lemma 3.3 we have that (q

, . . . , q

) is a regular element.

Now we determine regular elements with respect to the operation +. We need the following lemma:

Lemma 3.5. Let s, t ∈ W

(X

). Then

S

(s, t, t, . . . , t) = t if and only if s = x

for some 1 ≤ i ≤ n.

(That means, s + t = t if and only if s = x

for some 1 ≤ i ≤ n).

P roof. If s = x

for some 1 ≤ i ≤ n then S

(s, t, t, . . . , t) = t. Now we assume S

(s, t, t, . . . , t) = t and s / ∈ X

. Then op(s) ≥ 1 and we get

op(t) = op(S

(s, t, t, . . . , t))

=

n

X

j=1

vb

(s)op(t) + op(s)

≥ op(t) + op(s)

> op(t) which is a contradiction. Hence s ∈ X

.

From Theorem 3.4 using the embedding described in section 1, for regular elements with respect to + we obtain the following result:

Corollary 3.6. A term t ∈ W

(X

) is regular with respect to + if and only if it is idempotent.

By definition of the operation ∗ on W

(X

)

a vector (q

, . . . , q

) is v-regular if and only if there exists an element s ∈ W

(X

) such that

q

= S

(S

(q

, s, . . . , s), q

, . . . , q

) for 1 ≤ i ≤ n. By (C1) this means that

q

= S

(q

, S

(s, q

, . . . , q

), . . . , S

(s, q

, . . . , q

)),

for 1 ≤ i ≤ n. By Lemma 2.1 this is satisfied if and only if there is a j with 1 ≤ j ≤ n such that q

= x

for all 1 ≤ i ≤ n. Therefore, (q

, . . . , q

) is v-regular if and only if there is an integer j with 1 ≤ j ≤ n such that q

= x

for every i with 1 ≤ i ≤ n.

By Lemma 3.5 the term t ∈ W

(X

) is weakly regular if and only if there exists an integer i with 1 ≤ i ≤ n and terms s

, . . . , s

such that x

= S

(t, s

, . . . s

). By Lemma 2.2 this is satisfied if and only if there is an integer j for 1 ≤ j ≤ n such that t = x

and s

= x

. Altogether, this means that with respect to the binary operation + on W

(X

) the concepts of regular and of weakly regular elements are equal.

Now we consider regularity of terms of type τ

with respect to the op- erations ∗

and +

, respectively, derived from the generalized superposition operation S

.

Let X

:= X \ X

. If q

, . . . , q

∈ W

(X

), then for any s

, . . . s

∈ W

(X)

S

(S

(q

, s

, . . . , s

), q

, . . . , q

)

= S

(q

, q

, . . . q

)

= q

for 1 ≤ i ≤ n,

i.e. every (q

, . . . , q

) ∈ W

(X

)

is regular with respect to ∗

. If t ∈ W

(X

), then for every s ∈ W

(X) we have

S

(S

(t, s, . . . , s), t, . . . , t) = t.

Therefore, every t ∈ W

(X

) is regular with respect to +

. This gives the following results:

Theorem 3.7. Let q

, . . . , q

∈ W

(X). Then (q

, . . . , q

) is a regular vector in (W

(X)

; ∗

) if and only if for any i, j ∈ {1, 2, . . . , n} we have:

if x

∈ var(q

) then there exists l ∈ {1, 2, . . . , n} such that q

= x

.

Theorem 3.8. A term t ∈ W

(X) is regular with respect to +

if and only

if it is idempotent.

4. Ideals in menger algebras and Green’s relations In the next sections we will study Green’s relations in Menger algebras of terms. Green’s relations are studied for semigroups or for monoids. There- fore, it is quite natural to study Green’s relations for (W

(X

); +) and for (W

(X); +

). For Menger algebras of rank n in [7] Green’s relations were defined by using of ideals.

In [9], different kinds of ideals in Menger algebras of rank n were defined as follows:

Definition 4.1. Let (M ; S

) be a Menger algebra of rank n. A nonempty subset H of M is called an s-ideal, if from h ∈ H there follows S

(h, t

, . . . , t

) ∈ H for all t

, . . . , t

∈ M. A set H is called a v-ideal, if from h

, . . . , h

∈ H there follows S

(t, h

, . . . , h

) ∈ H for all t ∈ M . The set H is called an l-ideal, if at least one of t, h

, . . . , h

belongs to H, then S

(t, h

, . . . , h

) belongs to H.

Definition 4.2. Let (M ; S

) be a Menger algebra of rank n and let a, b ∈ M . (i) aLb if either a = b or if there are elements s

, . . . , s

, t

, . . . , t

∈ M

such that S

(a, s

, . . . , s

) = b and S

(b, t

, . . . , t

) = a.

(ii) aRb if either a = b or if there are elements s, t ∈ W

(X

) such that S

(s, a, a, . . . , a) = b and S

(t, b, b, . . . , b) = a.

(iii) D = R ◦ L(= L ◦ R).

(iv) H = R ∩ L.

(v) aJ b if either a = b or if there are elements s, s

, . . . , s

∈ W

(X

) with a = S

(s, s

, . . . , s

) such that at least one of the factors is equal to b and there are elements t, t

, . . . , t

∈ M with b = S

(t, t

, . . . , t

) such that at least one of the factors is equal to a.

It can be proved that (a, b) ∈ L if and only if a and b generate the same s-ideal, (a, b) ∈ R if and only if a and b generate the same v-ideal, and also (a, b) ∈ J if and only if a and b generate the same l-ideal; L and R are commuting equivalence relations ([9]).

Let 4

be the set of all pairs (s, s) of terms from W

(X

) (the

diagonal of W

(X

)).

Theorem 4.3. Let a, b ∈ W

(X

). Then aRb iff a = b, i.e., R = 4

.

P roof. If a = b, then by definition of R we have aRb. Assume that aRb. Then there exist s, t ∈ W

(X

) such that a = S

(s, b, b, . . . , b) and b = S

(t, a, a, . . . , a). This implies

a = S

(s, S

(t, a, a, . . . , a), S

(t, a, a, . . . , a), . . . , S

(t, a, a, . . . , a)) and then

a = S

(S

(s, t, t, . . . , t), a, a, . . . , a) by (C1).

By Lemma 3.3 and 2.2, we have S

(s, t, . . . , t) = x

for some 1 ≤ i ≤ n and so s = x

and t = x

for some 1 ≤ j ≤ n. We obtain a = S

(x

, b, . . . , b) = b.

Then we get:

Corollary 4.4. H = R and L = D.

For Green’s relation L we have:

Theorem 4.5. Let a, b ∈ W

(X

). Then aLb if and only if there exists a permutation r on the set {1, 2, . . . , n} such that b = S

(a, x

, x

, . . . , x

).

P roof. Assume that there exists a permutation r on {1, 2, . . . , n} such that b = S

(a, x

, x

, . . . , x

). Then

S

(b, x

, x

, . . . , x

)

= S

(S

(a, x

, x

, . . . , x

), x

, x

, . . . , x

)

= S

(a, S

(x

, x

, x

, . . . , x

), . . . , S

(x

, x

, x

, . . . , x

)) (by (C1))

= S

(a, x

, . . . , x

)

= S

(a, x

, x

, . . . , x

) = a.

Therefore aLb.

Conversely, assume that aLb. Then there exist elements t

, . . . , t

and s

, . . . , s

∈ W

(X

) such that S

(a, t

, . . . , t

) = b and S

(b, s

, . . . , s

) = a. Then we have:

S

(S

(a, t

, . . . , t

), s

, . . . , s

) = a

⇒ S

(a, S

(t

, s

, . . . , s

), . . . , S

(t

, s

, . . . , s

)) = a (by (C1)).

If var(a) = {x

, x

, . . . , x

} then S

(t

, s

, . . . , s

) = x

for all j = 1, 2, . . . , k and then for each i

there exists an l

such that t

= x

and s

= x

.

We notice that for 1 ≤ p, q ≤ k we have: if l

= l

then i

= i

because of s

= s

⇒ x

= x

⇒ i

= i

.

Therefore |{i

, i

, . . . , i

}| = |{l

, l

, . . . , l

}| implies that there exists a permutation r on the set {1, 2, . . . , n} such that r(i

) = l

for all 1 ≤ j ≤ k.

Then we have

b = S

(a, t

, . . . , t

, . . . , t

, . . . , t

, . . . , t

)

= S

(a, t

, . . . , x

, . . . , x

, . . . , x

, . . . , t

)

= S

(a, t

, . . . , x

, . . . , x

, . . . , x

, . . . , t

)

= S

(a, x

, . . . , x

, . . . , x

, . . . , x

, . . . , x

).

(Since x

∈ var(a) if p / / ∈ {i

, i

, . . . , i

}).

Then for Green’s relation J we have:

Theorem 4.6.

(i) If n = 1 then J = 4

.

(ii) If n ≥ 2 then J = W

(X

) × W

(X

).

P roof.

(i) Let a, b ∈ W

(X

) such that aJ b. Then we have the following four cases.

(1) a = S

(b, t) and b = S

(a, s) for some t, s ∈ W

(X

).

(2) a = S

(b, t) and b = S

(s, a) for some t, s ∈ W

(X

).

(3) a = S

(t, b) and b = S

(a, s) for some t, s ∈ W

(X

).

(4) a = S

(t, b) and b = S

(s, a) for some t, s ∈ W

(X

).

In all cases we obtain op(a) = op(b) + op(t) and op(b) = op(a) + op(s). These imply op(a) = op(a) + op(s) + op(t) and then op(s) = op(t) = 0 and hence s = x

= t. Thus, we have a = b in all four cases.

(ii) Let a, b ∈ W

(X

) where n ≥ 2. Then

a = S

(x

, a, b, b, . . . , b) and b = S

(x

, b, a, a, . . . , a).

This means aJ b for all a, b, ∈ W

(X

). Hence J = W

(X

) × W

(X

).

Now for a type τ

we consider the generalized superposition operation S

and the Menger algebra clone

τ

with infinitely many nullary operations.

Corresponding to Definition 4.2 we define Green’s relations L

, R

, D

, H

and J

. Let X

= X \ X

. For Green’s relation L

we have:

Theorem 4.7. Let a, b ∈ W

(X). Then aL

b if and only if there exists a permutation r on the set {1, 2, . . . , n} such that b = S

(a, x

, . . . , x

).

P roof. By definition we have aL

b if either a = b or if there are ele- ments s

, . . . , s

, t

, . . . , t

∈ W

(X) such that S

(a, s

, . . . , s

) = b and S

(b, t

, . . . , t

) = a.

We consider the following cases:

1. var(a) or var(b) ⊆ X

: In this case from aL

b we get b = S

(a, s

, . . . , s

) = a.

2. var(a), var(b) 6⊆ X

: In this case we have var(a) ∩ X

6= ∅ and

var(b) ∩ X

6= ∅. Now we proceed as in the proof of Theorem 4.5.

For R

we have:

Theorem 4.8.

R

= W

(X

)

∪ 4

.

P roof. By definition we have aR

b if either a = b or if there are elements s, t ∈ W

(X) such that S

(s, a, . . . , a) = b and S

(t, b, . . . , b) = a. We consider again the following cases:

1. var(a) or var(b) ⊆ X

: If var(a) ⊆ X

, then var(S

(s, a, . . . , a)) = var(b) ⊆ X

and conversely, if var(b) ⊆ X

, then var(a) ⊆ X

. So, we may assume that both, var(a) and var(b) are subsets of X

. Let R

|

⊆ W

(X

)

be the restriction of R

to W

(X

). If var(a), var(b) ⊆ X

, then S

(b, a, . . . , a) = b and S

(a, b, . . . , b) = a and there- fore aL

b. This shows R

|

= W

(X

)

.

2. var(a) 6⊆ X

and var(b) 6⊆ X

: In this case we have var(s) ∩ X

6= ∅, var(t) ∩ X

6= ∅, var(S

(s, t, . . . , t)) ∩ X

6= ∅ and var(S

(t, s, . . . , s)) ∩ X

6= ∅. From aR

b we obtain by substitution and by (Cg1), S

(S

(s, t, . . . , t), b, . . . , b) = b and S

(S

(t, s, . . . , s), a, . . . , a) = a. Similar to Lemma 3.5 there follows

S

(s, t, . . . , t) = x

, S

(t, s, . . . , s) = x

for some 1 ≤ i, j ≤ n. But then similar as in Lemma 2.2 we conclude that s = x

, t = x

and t = x

, s = x

, i.e. s = t = x

. Then we have a = b. The converse is clear.

For Green’s relation J

we have:

Theorem 4.9. If n = 1, then J

= W

(X

) × W

(X

) ∪ 4

. P roof. We consider the following two cases:

1. var(a) ⊆ X

or var(b) ⊆ X

: In this case from var(a) ⊆ X

we have a = S

(a, b) and then var(b) ⊆ X

and from var(b) ⊆ X

and b = S

(b, a), we obtain var(a) ⊆ X

. Therefore in this case we get J

|W

(X

)

= W

(X

)

.

2. var(a) ∩ X

6= ∅ and var(b) ∩ X

6= ∅: Assume that a 6= b and aJ

b,

then the following cases are possible:

(1) a = S

(b, t) and b = S

(a, s) for some t, s ∈ W

(X).

(2) a = S

(b, t) and b = S

(s, a) for some t, s ∈ W

(X).

(3) a = S

(t, b) and b = S

(a, s) for some t, s ∈ W

(X).

(4) a = S

(t, b) and b = S

(s, a) for some t, s ∈ W

(X).

Since in the first case a = S

(b, t) and var(a)∩X

6= ∅ we have var(t)∩X

6=

∅ and similarly we have var(s) ∩ X

6= ∅. In the other three cases one has also var(t) ∩ X

6= ∅ and var(s) ∩ X

6= ∅. Using the formula

op(S

(q, t)) = op(t) + op(q), q, t ∈ W

(X)

it is not difficult to see that for arbitrary terms u, v, w ∈ W

(X) from u = S

(v, w), var(u) ∩ X

6= ∅ and var(v) ∩ X

6= ∅ there follows op(u) ≥ op(v)+op(w). Using this inequality we obtain in all four cases op(s) = 0 and op(t) = 0 and thus by var(t)∩X

6= ∅, var(s)∩X

6= ∅ we get s = t = x

and then a = b, a contradiction. Altogether we have J

= W

(X

)

∪ 4

.

For n ≥ 2 we obtain

Theorem 4.10. If n ≥ 2, then J

= W

(X) × W

(X).

P roof. Let (a, b) ∈ W

(X)

. Then

a = S

(x

, a, b, . . . , b) and b = S

(x

, b, a, . . . , a).

This means aJ

b.

Moreover, we have H

= R

∩ L

= 4

and D

= R

∨ L

= W

(X

)

∨ L

. For D

we have even:

Corollary 4.11. D

= R

∪ L

= W

(X

)

∨ L

.

P roof. The inclusion R

∪ L

⊆ D

is clear. Let (a, b) ∈ D

. By

D

= L

◦ R

, there is a term c ∈ W

(X) such that (a, c) ∈ R

and

(c, b) ∈ L

. We consider the following two cases:

1. var(a) ⊆ X

or var(b) ⊆ X

: Since (c, b) ∈ L

, by Theorem 4.7 there is a permutation r on {1, 2, . . . , n} such that b = S

(c, x

, . . . , x

). If var(b) ⊆ X

, we get var(c) ⊆ X

and b = c. It follows (a, b) = (a, c) ∈ R

and var(a) ⊆ X

. If var(a) ⊆ X

, then (a, c) ∈ R

implies var(c) ⊆ X

and then by (c, b) ∈ L

also var(b) ⊆ X

and we continue as we did in the first case. This shows also

var(a) ⊆ X

or var(b) ⊆ X

⇔ var(a) ⊆ X

and var(b) ⊆ X

. 2. var(a) ∩ X

6= ∅ and var(b) ∩ X

6= ∅: Then by Theorem 4.8 we get a = c

and this implies (a, b) = (c, b) ∈ L

. Altogether we have D

⊆ R

∪ L

.

5. Green’s Relations on (W

(X

); +)

Now we consider Green’s relations with respect to the the semigroup (W

(X

); +) where + is defined by a+b := S

(a, b, b . . . , b). Let (W

(X

))

be the monoid arising from (W

(X

); +) by adding a neutral element 0.

Corresponding to the usual definition for Green’s relations L

, R

, D

, J

, and H

we have:

aR

b :⇔ a = b or a = S

(b, s, s, . . . , s) and

b = S

(a, t, t, . . . , t) for some s, t ∈ W

(X

).

aL

b : ⇔ a = b or a = S

(s, b, b, . . . , b) and

b = S

(t, a, a, . . . , a) for some s, t ∈ W

(X

).

aJ

b : ⇔ a = S

(s, S

(b, t, t, . . . , t), . . . , S

(b, t, t, . . . , t)) and b = S

(s

, S

(a, t

, t

, . . . , t

), . . . , S

(a, t

, t

, . . . , t

))

for some s, s

, t, t

∈ (W

(X

))

. H

= L

∩ R

and D

= R

◦ L

(= L

◦ R

).

By definition and Theorem 4.3 we have L

= R = 4

, H

= L

,

and D

= R

. It is left to determine R

and J

.

Theorem 5.1. Let a, b ∈ W

(X

). Then

aR

b :⇔ a = b or a = S

(b, x

, x

, . . . , x

) and b = S

(a, x

, x

, . . . , x

) for some i, j ∈ {1, 2, . . . , n}.

P roof. Assume that aR

b. If a 6= b then a = S

(b, s, s, . . . , s) and b = S

(a, t, t, . . . , t) for some s, t ∈ W

(X

), so we have

a = S

(b, s, s, . . . , s)

= S

(S

(a, t, t, . . . , t), s, . . . , s)

= S

(a, S

(t, s, s, . . . , s), . . . , S

(t, s, s, . . . , s)) (by (C1)).

By Lemma 2.2, we get S

(t, s, . . . , s) = x

for some j and then var(a) = {x

} because of a = S

(a, x

, . . . , x

). Since S

(t, s, . . . , s) = x

, by Lemma 2.2 we have t = x

and s = x

for some 1 ≤ i ≤ n. Then b = S

(a, t, . . . , t) = S

(a, x

, . . . , x

) and a = S

(b, x

, . . . , x

).

Conversely, if a = b or a = S

(b, x

, . . . , x

) and b = S

(a, x

, . . . , x

) for some i, j ∈ {1, 2, . . . , n}, then by definition of R

we have aR

b.

This means, if aR

b and a 6= b, then var(a) = x

and var(b) = x

and a arises from b by exchanging x

and x

.

For J

we have:

Theorem 5.2. J

= R

.

P roof. We will show that J

⊆ R

. Let a, b ∈ W

(X

) such that aJ

b.

Then a = s + b + t and b = s

+ a + t

for some s, s

, t, t

∈ (W

(X

))

. So we have

op(a) = op(s + b + t)

= op(S

(s, S

(b, t, . . . , t), . . . , S

(b, t, . . . , t))

≥ op(s) + op(S

(b, t, . . . , t))

≥ op(s) + op(b) + op(t)

= op(s) + op(s

+ a + t

) + op(t)

≥ op(s) + op(s

) + op(a) + op(t

) + op(t).

It follows that op(s) + op(s

) + op(t

) + op(t) = 0, and then s, s

, t, t

are variables or the neutral element of the monoid (W

(X

))

. It is not difficult to see that in all of the cases we get

a = b + t and b = a + t

. This means aR

b. Since R

⊆ J

, we get J

= R

.

Altogether we have H

= L

= 4

, R

= D

= J

.

If we consider only unary terms, then S

= +. In this case, all of Green’s relations are equal.

Proposition 5.3. In the diagonal Menger algebra (W

(X

); +) (in the monoid (W

(x

); S

, x

)) , we have

H

= L

= R

= D

= J

= 4

.

P roof. It is enough to show that J ⊆ 4

. Let a, b ∈ W

(X

) and aJ b. Then there are elements s, t, s

, t

∈ (W

(X

))

such that a = s + b + t and b = s

+ a + t

. Consider

a = s + b + t

= S

(s, S

(b, t))

= S

(s, S

(s

+ a + t

, t))

= S

(s, S

(S

(s

, S

(a, t

)), t)).

Then op(a) = op(s) + op(s

) + op(a) + op(t

) + op(t).

This implies op(s) = op(s

) = op(t

) = op(t) = 0, and so s = s

= t = t

= x

. Since x

is a neutral element of (W

(X

); +), we get a = x

+ b + x

= b. Hence J = 4

.

Now for a type τ

we consider the generalized superposition operation S

and the diagonal algebra (W

(X); +

), where a +

b := S

(a, b, . . . , b).

Then we define Green’s relations L

, R

, D

, H

and J

in the usual way.

Let X

= X \ X

. For Green’s relation R

we have:

Theorem 5.4. Let a, b ∈ W

(X). Then

aR

b :⇔ a = b or a = S

(b, x

, x

, . . . , x

) and b = S

(a, x

, x

, . . . , x

) for some i, j ∈ {1, 2, . . . , n}.

P roof. We consider the following two cases:

1. var(a) ⊆ X

or var(b) ⊆ X

: If var(a) ⊆ X

, then S

(a, t, . . . , t) = a for arbitrary t ∈ W

(X) and if var(b) ⊆ X

, then S

(b, s, . . . , s) = b for arbitrary s ∈ W

(X) . This shows that R

|

= 4

. 2. var(a) ∩ X

6= X

and var(b) ∩ X

6= X

: If aR

b and a 6= b, then

a = S

(b, s, . . . , s) and b = S

(a, t, . . . , t) for some s, t ∈ W

(X). Then we obtain

a = S

(b, s, . . . , s)

= S

(S

(a, t, . . . , t), s, . . . , s)

= S

(a, S

(t, s, . . . , s), . . . , S

(t, s, . . . , s)) by (Cg1).

By var(a) ∩ X

6= ∅, similar to Lemma 2.1 we obtain S

(t, s, . . . , s) = x

for x

∈ var(a) ∩ X

and then similar as in Lemma 2.2 we get t = x

and s = x

for some 1 ≤ j ≤ n.

By definition and Theorem 4.8 we have L

= R

= W

(X

)

∪ 4

and then H

= R

∩ L

= 4

. For D

we get D

= R

∨ L

= R

∨ W

(X

)

. It is left to determine J

.

Theorem 5.5.

J

= R

∪ L

= R

∪ W

(X

)

.

P roof. Let a, b, a 6= b ∈ W

(X) such that aJ

b. Then a = s +

b +

t

and b = s

+

a+

t

for some s, s

, t, t

∈ W

(X)

. We consider the following

two cases:

1. var(a) ⊆ X

or var(b) ⊆ X

: Then a +

t

= a and therefore b = s

+

a.

Similarly, from var(b) ⊆ X

there follows a = s +

b. Now we show that var(a) ⊆ X

if and only if var(b) ⊆ X

. Indeed, if var(a) ⊆ X

, and if we assume that var(s

) ⊆ X

, then s

+

a = s

and b = s

+

a implies b = s

and thus var(b) ⊆ X

. If var(s

) ∩ X

6= ∅, then from b = s

+

a we obtain

var(b) = var(a) ∪ (var(s

) ∩ X

) ⊆ X

∪ X

= X

.

Similarly, from var(b) ⊆ X

there follows var(a) ⊆ X

. This means, that var(a) ⊆ X

or var(b) ⊆ X

implies a = s +

b and b = s

+

a for some s, s

∈ W

(X) and then aL

b. From L

|

= W

(X

)

we obtain J

|

= W

(X

)

.

2. var(a)∩X

6= ∅ and var(b)∩X

6= ∅: From a = s+

b+

t we get var(s)∩

X

6= ∅, since from var(s) ⊆ X

, we obtained a = s and so var(a) ⊆ X

, a contradiction. In a similar way we show that var(s

) ∩ X

6= ∅. The next step is to show that var(t) ∩ X

6= ∅ and var(t

) ∩ X

6= ∅. Indeed, if var(t) ∩ X

= ∅, then var(t) ⊆ X

and then

var(a) ⊆ var(t) ∪ (var(s +

b) ∩ X

) ⊆ X

∪ X

= X

,

a contradiction. This proves var(t) ∩ X

6= ∅. Similarly, we show that var(t

) ∩ X

6= ∅.

Using the formula

op(S

(q, t

, . . . , t

))

= P

k=1

vb

(q)op(t

) + op(q), q, t

, . . . , t

∈ W

(X)

it is not difficult to see that for arbitrary terms u, v, w ∈ W

(X) from

u = v +

w and var(u) ∩ X

6= ∅ and var(v) ∩ X

6= ∅ there follows

op(u) ≥ op(v) + op(w). Using this inequality we obtain

op(a) = op(s +

b +

t)

= op(S

(s, S

(b, t, . . . , t), . . . , S

(b, t, . . . , t))

≥ op(s) + op(b) + op(t)

= op(s) + op(s

+

a +

t

) + op(t)

≥ op(s) + op(s

) + op(a) + op(t

) + op(t).

It follows that op(s) + op(s

) + op(t

) + op(t) = 0, and then s, s

, t, t

are variables or the neutral element of the monoid (W

(X

))

. It is not difficult to see that in all of the cases we get

a = b +

t and b = a +

t

,

i.e. aR

b and because of R

⊆ J

we have J

= R

in the second case and together with the first case we obtain J

= R

∪ W

(X

)

.

As a final result we get L

= D

= J

= 4

∪ W

(X

)

, H

= 4

and R

as in Theorem 5.4.

References

[1] K. Denecke, Stongly Solid Varieties and Free Generalized Clones, Kyungpook Math. J. 45 (2005), 33–43.

[2] K. Denecke and S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D.C., 2002.

[3] K. Denecke and S.L. Wismath, Complexity of Terms, Composition and Hypersubstitution, Int. J. Math. Math. Sci. 15 (2003), 959–969.

[4] K. Denecke and P. Jampachon, N-solid varieties and free Menger algebras of rank n, East-West Journal of Mathematics 5 (1) (2003), 81–88.

[5] K. Denecke and P. Jampachon, Clones of Full Terms, Algebra Discrete Math.

4 (2004), 1–11.

[6] K. Denecke and J. Koppitz, M-solid Varieties of Algebras, Advances in

Mathematics, Springer Science+Business Media, Inc., 2006.

[7] J.M. Howie, Fundamenntals of Semigroup Theory, Oxford Science Publica- tions, Clarendon Press, Oxford 1995.

[8] K. Menger, The algebra of functions: past, present, future, Rend. Mat. 20 (1961), 409–430.

[9] B.M. Schein and V.S. Trohimenko, Algebras of multiplace functions, Semi- group Forum 17 (1979), 1–64.

[10] V.S. Trohimenko, v-regular Menger algebras, Algebra Univers. 38 (1997), 150–164.

Received July 2005

Revised August 2005