WattapongPuninagoolandSorasakLeeratanavalee τ =( n ) THEMONOIDOFGENERALIZEDHYPERSUBSTITUTIONSOFTYPE DiscussionesMathematicaeGeneralAlgebraandApplications30 ( 2010 ) 173–191

THE MONOID OF GENERALIZED HYPERSUBSTITUTIONS OF TYPE τ = (n)

Wattapong Puninagool and

Sorasak Leeratanavalee

Department of Mathematics, Faculty of Science, Chiang Mai University,

Chiang Mai 50200, Thailand e-mail: wattapong1p@yahoo.com e-mail: scislrtt@chiangmai.ac.th

Abstract

A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green’s relations, have been studied for type (n) by S.L. Wismath.

A generalized hypersubstitution of type τ = (n) is a mapping σ which takes the n-ary operation symbol f to a term σ(f ) which does not necessarily preserve the arity. Any such σ can be inductively ex- tended to a map ˆ σ on the set of all terms of type τ = (n), and any two such extensions can be composed in a natural way. Thus, the set Hyp

(n) of all generalized hypersubstitutions of type τ = (n) forms a monoid. In this paper we study the semigroup properties of Hyp

(n).

∗

Corresponding author.

In particular, we characterize the idempotent and regular generalized hypersubstitutions, and describe some classes under Green’s relations of this monoid.

Keywords: monoid, regular elements, idempotent elements, Green’s relations, generalized hypersubstitution.

2000 Mathematics Subject Classification: 20M05, 20M99, 20N02.

1. Introduction

Identities are used to classify algebras into collections called varieties. Hy- peridentities are used to classify varieties into collections called hypervari- eties. The concepts of hyperidentities and hypervarieties were introduced by W. Taylor in 1981 [7]. Hyperidentities in a variety V are identities which have the property that, after replacing the operation symbols which occur in these identities by any terms of the same arity, the resulting equation is still satisfied in the variety. The main tool to study hyperidentities is the concept of a hypersubstitution, which was introduced by K. Denecke, D.

Lau, R. P¨ oschel and D. Schweigert in 1991 [1]. Let τ = (n

)

be a type and let W

(X) be the set of all terms of type τ built up by operation symbols from {f

|i ∈ I} where f

is n

-ary and variables from a countably infinite alphabet X := {x

, x

, . . .}. A hypersubstitution of type τ is a mapping σ : {f

|i ∈ I} → W

(X) which maps n

-ary operation symbols to n

-ary terms. Let Hyp(τ ) be the set of all hypersubstitutions of type τ . For every σ ∈ Hyp(τ ) induces a mapping ˆ σ : W

(X) → W

(X) as follows: for any t ∈ W

(X), ˆ σ[t] is defined inductively by

(i) ˆ σ[x] := x ∈ X,

(ii) ˆ σ[f

(t

, . . . , t

)] := σ(f

)(ˆ σ[t

], . . . , ˆ σ[t

]), for any n

-ary operation symbol f

.

It turns out that (Hyp(τ ); ◦

, σ

) is a monoid where σ

◦

σ

:= ˆ σ

◦ σ

and σ

(f

) = f

(x

, . . . , x

) is the identity element.

In 2000, S. Leeratanavalee and K. Denecke generalized the concept of a

hypersubstitution to a generalized hypersubstitution [2]. S. Leeratanavalee

and K. Denecke used generalized hypersubstitutions as the tools to study

strong hyperidentities and used strong hyperidentities to classify varieties into collections called strong hypervarieties. Varieties whose identities are closed under arbitrary application of generalized hypersubstitutions are called strongly solid.

A generalized hypersubstitution of type τ , or for short simply a gener- alized hypersubstitution, is a mapping σ which maps each n

-ary operation symbol of type τ to a term of this type in W

(X) which does not necessarily preserve the arity. We denoted the set of all generalized hypersubstitutions of type τ by Hyp

(τ ). First, we define inductively the concept of generalized superposition of terms S

: W

(X)

→ W

(X) by the following steps:

(i) If t = x

, 1 ≤ j ≤ m, then S

(x

, t

, . . . , t

) := t

. (ii) If t = x

, m < j ∈ IN, 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

)).

To define a binary operation on Hyp

(τ ), we extend a generalized hyper- substitution σ to a mapping ˆ σ : W

(X) → W

(X) inductively defined as follows:

(i) ˆ σ[x] := x ∈ X,

(ii) ˆ σ[f

(t

, . . . , t

)] := S

(σ(f

), ˆ σ[t

], . . . , ˆ σ[t

]), for any n

-ary opera- tion symbol f

.

Then we define a binary operation ◦

on Hyp

(τ ) by σ

◦

σ

:= ˆ σ

◦ σ

where ◦ denotes the usual composition of mappings and σ

, σ

∈ Hyp

(τ ).

Let σ

be the hypersubstitution which maps each n

-ary operation symbol f

to the term f

(x

, . . . , x

). S. Leeratanavalee and K. Denecke proved the following propositions.

Proposition 1.1 ([2]). For arbitrary terms t, t

, . . . , t

∈ W

(X) and for arbitrary generalized hypersubstitutions σ, σ

, σ

we have

(i) S

(ˆ σ[t], ˆ σ[t

], . . . , ˆ σ[t

]) = ˆ σ[S

(t, t

, . . . , t

)],

(ii) (ˆ σ

◦ σ

)ˆ= ˆ σ

◦ ˆ σ

.

Proposition 1.2 ([2]). Hyp

(τ ) = (Hyp

(τ ); ◦

, σ

) is a monoid, with σ

as the identity element, and the set of all hypersubstitutions of type τ forms a submonoid of Hyp

(τ ).

Many properties of the monoid of hypersubstitutions of type τ = (n) were described by S.L. Wismath [8]. In this paper we extend the results from [8]

to the case of Hyp

(n).

2. Projection and dual generalized hypersubstitutions of type τ = (n)

We assume that from now on the type τ = (n), for some n ∈ IN, i.e. we have only one n-ary operation symbol, say f . By σ

we denote the generalized hypersubstitution which maps f to the term t in W

(X). A generalized hypersubstitution σ

is called a projection generalized hypersubstitution if t is a variable [3]. We denoted the set of all projection generalized hypersub- stitutions of type τ = (n) by P

(n), i.e. P

(n) = {σ

|x

∈ X}.

Lemma 2.1. For any σ

∈ Hyp

(n) and σ

∈ P

(n), we have (i) σ

◦

σ

= σ

,

(ii) σ

◦

σ

∈ P

(n).

P roof. (i) We have (σ

◦

σ

)(f ) = (ˆ σ

◦ σ

)(f ) = ˆ σ

[σ

(f )] = ˆ σ

[x

] = x

= σ

(f ). So σ

◦

σ

= σ

.

(ii) We will proceed by induction on the complexity of the term t. If t ∈ X, then by (i) we get σ

◦

σ

= σ

∈ P

(n). Assume that t = f (u

, . . . , u

) and σ

◦

σ

, . . . , σ

◦

σ

∈ P

(n). Thus ˆ σ

[u

], . . . , ˆ σ

[u

] ∈ X. We have (σ

◦

σ

)(f ) = (σ

◦

σ

)(f ) = S

(x

, ˆ σ

[u

], . . . , ˆ σ

[u

]).

If x

∈ X

where X

= {x

, . . . , x

}, then (σ

◦

σ

)(f ) = ˆ σ

[u

] ∈ X. If i > n, then (σ

◦

σ

)(f ) = x

∈ X. So σ

◦

σ

∈ P

(n).

Corollary 2.2.

(i) P

(n) ∪ {σ

} is a submonoid of Hyp

(n) and P

(n) is the small-

est two-sided ideal of Hyp

(n), called the kernel of Hyp

(n). Thus,

Hyp

(n) is not simple.

(ii) P

(n) is the set of all right-zero elements of Hyp

(n), so that P

(n) itself is a right-zero band.

(iii) Hyp

(n) contains no left-zero elements.

P roof. These follow immediately from Lemma 2.1.

Another special kind of generalized hypersubstitutions in Hyp

(n) are dual generalized hypersubstitutions, which are defined using permutations of the set J := {1, . . . , n}. For any such permutation π, we let σ

= σ

. We let D

be the set of all such dual generalized hyper- substitutions.

Lemma 2.3.

(i) For any two permutations π and ρ, σ

◦

σ

= σ

.

(ii) For any permutation π with the inverse permutation π

, the general- ized hypersubstitutions σ

and σ

are inverse of each other.

P roof. (i) We have (σ

◦

σ

)(f ) = ˆ σ

[f (x

, . . . , x

)] = S

(f (x

, . . . , x

), x

, . . . , x

) = f (x

, . . . , x

) = σ

(f ).

(ii) This follows from (i).

Lemma 2.4. If σ ◦

ρ ∈ D

, then both σ and ρ are in D

.

P roof. Let σ(f ) = f (u

, . . . , u

) and ρ(f ) = f (v

, . . . , v

). Since σ ◦

ρ ∈ D

, thus there exists a permutation π such that (σ ◦

ρ)(f ) = f (x

, . . . , x

). So f (x

, . . . , x

) = (σ ◦

ρ)(f ) = S

(f (u

, . . . , u

), ˆ

σ[v

], . . . , ˆ σ[v

]). Since π is a permutation, thus this forces all the u

’s to be distinct variables in X

, and all the v

’s to be distinct variables in X

. It follows that both σ and ρ are in D

.

Corollary 2.5. D

is a submonoid of Hyp

(n) which forms a group, and

no other elements of Hyp

(n) have inverses in Hyp

(n). Thus, D

is the

largest subgroup of Hyp

(n).

Lemma 2.6. Let F be the set of generalized hypersubstitutions of the form σ

for i ∈ IN. Let M = P

(n) ∪ D

∪ F . Then M is a submonoid of Hyp

(n).

P roof. It is straightforward to check that any product of two elements in M is also in M .

3. Idempotent and regular elements in Hyp G (n)

All idempotent elements of the monoid of all generalized hypersubstitutions of type τ = (2) were studied by W. Puninagool and S. Leeratanavalee [6]

and all regular elements of the monoid of all generalized hypersubstitutions of type τ = (2) were studied by W. Puninagool and S. Leeratanavalee [4].

In this section, we characterize the idempotent and regular generalized hy- persubstitutions of type τ = (n).

We know from Corollary 2.2 (ii) that every projection generalized hy- persubstitution is idempotent. We let G(n) := {σ

|t / ∈ X, var(t) ∩ X

= ∅}

where var(t) denotes the set of all variables occurring in t.

Lemma 3.1. If σ

∈ G(n) and σ

∈ Hyp

(n) \ P

(n), then σ

◦

σ

= σ

, i.e. G(n) itself is a left zero band.

P roof. Let s = f (v

, . . . , v

). We have (σ

◦

σ

)(f ) = S

(t, ˆ σ

[v

], . . . , ˆ

σ

[v

]) = t since there is nothing to substitute in the term t. So σ

◦

σ

= σ

.

Then we consider only the case σ

∈ Hyp

(n) \ P

(n) and var(t) ∩ X

6= ∅.

Theorem 3.2. Let t = f (t

, . . . , t

) ∈ W

(X) and ∅ 6= var(t) ∩ X

= {x

, . . . , x

}. Then σ

is idempotent if and only t

= x

for all k ∈ {1, . . . , m}.

P roof. Assume that σ

is idempotent. Then S

(f (t

, . . . , t

), ˆ

σ

[t

], . . . , ˆ σ

[t

]) = σ

1,...,tn)

(f ) = σ

(f ) =

f (t

, . . . , t

). Suppose that there exists k ∈ {1, . . . , m} such that t

6= x

.

If t

∈ X, then ˆ σ

[t

] = t

6= x

. So S

(f (t

, . . . , t

), ˆ σ

[t

], . . . , ˆ σ

[t

]) 6= f (t

, . . . , t

) and we have a contradiction.

If t

∈ X, then ˆ / σ

[t

] / ∈ X. We obtain op(t) = op(S

(f (t

, . . . , t

), ˆ

σ

[t

], . . . , ˆ σ

[t

])) > op(t) where op(t) denotes the number of all operation symbols occurring in t. This is a contradiction.

The proof of the converse direction is straightforward.

Now, we characterize the regular generalized hypersubstitutions of type τ = (n). At first we want to recall the definition of the regular element.

Definition 3.3. An element a of a semigroup S is called regular if there exists x ∈ S such that axa = a. The semigroup S is called regular if all its elements are regular.

Lemma 3.4. Let t ∈ W

(X) and ∅ 6= var(t) ∩ X

= {x

, . . . , x

} and let a = f (a

, . . . , a

) ∈ W

(X). If ˆ σ

[a] = t, then a

= x

for all l = i

, . . . , i

.

P roof. Assume that ˆ σ

[a] = t. Then t = ˆ σ

[a] = S

(t, ˆ σ

[a

], . . . , ˆ σ

[a

]).

We will show that a

= x

for all l = i

, . . . , i

. Suppose that there exists j

∈ {i

, . . . , i

} such that a

6= x

. If a

= x

∈ X where x

6= x

, then ˆ σ

[a

] = x

. It follows that t 6= S

(t, ˆ σ

[a

], . . . , ˆ σ

[a

]).

This is a contradiction. If a

∈ X, then ˆ / σ

[a

] / ∈ X. It follows that op(t) = op(ˆ σ

[a]) = S

(t, ˆ σ

[a

], . . . , ˆ σ

[a

]) > op(t) and we have a contra- diction.

Theorem 3.5. Let t = f (t

, . . . , t

) ∈ W

(X) and ∅ 6= var(t) ∩ X

= {x

, . . . , x

}. Then σ

is regular if and only if there exist j

, . . . , j

∈ {1, . . . , n} such that t

= x

, . . . , t

= x

.

P roof. Assume that σ

is regular. Then there exists σ

∈ Hyp

(n) such that σ

◦

σ

◦

σ

= σ

. Since t / ∈ X, thus s / ∈ X. Then s = f (s

, . . . , s

) for some s

, . . . , s

∈ W

(X). From σ

◦

σ

◦

σ

= σ

, thus ˆ

σ

[ˆ σ

[t]] = t. By Lemma 3.4, ˆ σ

[t] = f (u

, . . . , u

) for some u

, . . . , u

∈ W

(X) where u

= x

, . . . , u

= x

. From ˆ σ

[t] = f (u

, . . . , u

), thus S

(f (s

, . . . , s

), ˆ σ

[t

], . . . , ˆ σ

[t

]) = f (u

, . . . , u

). Since u

= x

, . . . , u

= x

thus s

, . . . , s

∈ X

. Let s

= x

, . . . , s

= x

. Hence

t

= x

, . . . , t

= x

.

Conversely, assume the condition holds. Let s = f (s

, . . . , s

) ∈ W

(X) where s

, . . . , s

∈ W

(X) such that s

= x

, . . . , s

= x

.

Then (σ

◦

σ

◦

σ

)(f ) = ˆ σ

[ˆ σ

[t]] = ˆ σ

[S

(f (s

, . . . , s

), ˆ σ

[t

], . . . , ˆ

σ

[t

])] = ˆ σ

[f (k

, . . . , k

)](where k

= x

, . . . , k

= x

) = S

(t, ˆ σ

[k

], . . . , ˆ σ

[k

]) = t. Hence σ

◦

σ

◦

σ

= σ

.

4. Term properties of the composition operation

We need to know more about the result of the composing two generalized hypersubstitutions in Hyp

(n). In particular, we want to know how long the term corresponding to σ

◦

σ

is and what variables it uses, compared to the lengths of the terms s and t and the variables they use. We begin with the necessary definitions.

Definition 4.1. Let t ∈ W

(X). We define the length of t inductively by:

(i) The length of t is 1 if t is a variable.

(ii) If t is a compound term f (t

, . . . , t

), then the length of t is the sum of the lengths of the terms t

, . . . , t

.

(iii) This length counts the total number of variable occurences in the term t, and will be denoted by vb(t).

Definition 4.2 ([8]). Let t ∈ W

(X). We define some new terms, related to t, as follows. Recall that J := {1, . . . , n}.

(i) Let α be any function from J to J. C

[t] is the term formed from t by replacing each occurrence in t of a variable x

∈ X

by the variable x

i.e, C

[t] = S

(t, x

, . . . , x

).

(ii) Let π be any permutation of J. π[t] is the term defined induc- tively by π[x

] = x

for any variable x

, and π[f (u

, . . . , u

)] = f (π[u

], . . . , π[u

]).

The previous length results for the type τ = (2) were found by W. Puni-

nagool and S. Leeratanavalee in [6] and S.L. Wismath in [8]. The next two

lemmas show how these results can be generalized to the type τ = (n).

Lemma 4.3. Let n ∈ N with n > 1. Let σ

◦

σ

= σ

. Then w is a longer term than f (u

, . . . , u

), unless the terms f (u

, . . . , u

) and f (v

, . . . , v

) satisfy the following condition (Q):

(Q) If a variable x

∈ X

is used anywhere in the term f (u

, . . . , u

), then the entry v

in f (v

, . . . , v

) is a variable.

P roof. If var(f (u

, . . . , u

))∩X

= ∅, then f (u

, . . . , u

) and f (v

, . . . , v

) satisfy the condition (Q). Let var(f (u

, . . . , u

)) ∩ X

= {x

, . . . , x

}.

If v

∈ X for all j ∈ {1, . . . , k}, then f (u

, . . . , u

) and f (v

, . . . , v

) satisfy the condition (Q). If there exists j ∈ {1, . . . , k} where v

∈ X, / then ˆ σ

[v

] / ∈ X. Since n > 1 and ˆ σ

[v

] / ∈ X, thus vb(ˆ σ

[v

]) > 1. So vb(w) = vb(S

(f (u

, . . . , u

), ˆ σ

[v

], . . . , ˆ σ

[v

])) > vb(f (u

, . . . , u

)).

Lemma 4.4. Let σ

∈ Hyp

(n) where t / ∈ X and x

, . . . , x

∈ var(t). Then for any s ∈ W

(X), vb(ˆ σ

[s]) ≥ vb(s).

P roof. We will proceed by induction on the complexity of the term s. If s ∈ X, then vb(ˆ σ

[s]) = vb(s). Assume that s = f (u

, . . . , u

) and vb(ˆ σ

[u

]) ≥ vb(u

) for all 1 ≤ i ≤ n. Then vb(ˆ σ

[s]) = vb(S

(t, ˆ σ

[u

], . . . , ˆ σ

[u

])) ≥ vb(f (u

, . . . , u

)) since x

, . . . , x

∈ var(t) and vb(ˆ σ

[u

]) ≥ vb(u

) for all 1 ≤ i ≤ n.

Lemma 4.5. Let σ

◦

σ

= σ

where vb(f (u

, . . . , u

)) > n.

If x

, . . . , x

∈ var(f (u

, . . . , u

)), then w is a longer term than f (v

, . . . , v

).

P roof. We write σ = σ

. From σ

◦

σ

= σ

, thus we get w = S

(f (u

, . . . , u

), ˆ σ[v

], . . . , ˆ σ[v

]). Since x

, . . . , x

∈ var(f (u

, . . . , u

)), thus ˆ σ[v

] is used in w for all 1 ≤ i ≤ n. We will proceed by induction on the complexity of the term f (v

, . . . , v

). If v

, . . . , v

∈ X, then vb(w) = vb(f (u

, . . . , u

)) > n = vb(f (v

, . . . , v

)). Assume that the claim holds for any term of length not less than n but less than k, and f (v

, . . . , v

) has length k. Since vb(f (v

, . . . , v

)) = k > n, thus there exists i ∈ {1, . . . , n} such that vb(v

) ≥ n. By induction, we get vb(ˆ σ[v

]) > vb(v

).

By Lemma 4.4, any other v

has vb(ˆ σ[v

]) ≥ vb(v

). Since all the ˆ σ[v

] are

used in w for all 1 ≤ i ≤ n, thus w is longer than f (v

, . . . , v

).

Lemma 4.6. Let σ

, σ

∈ Hyp

(n).

(i) var((σ

◦

σ

)(f )) ∩ X

⊆ var(t) ∩ X

.

(ii) If s uses only one variable, then the term for σ

◦

σ

uses only one variable (not necessarily the same variable as s).

P roof. We will proceed by induction on the complexity of the term t.

(i) If t ∈ X, then (σ

◦

σ

)(f ) = t. So var((σ

◦

σ

)(f ))∩ X

⊆ var(t)∩

X

. Assume that t = f (t

, . . . , t

) and var(ˆ σ

[t

])∩X

⊆ var(t

)∩X

for all 1 ≤ i ≤ n. So var((σ

◦

σ

)(f )) ∩ X

= var(S

(s, ˆ σ

[t

], . . . , ˆ σ

[t

])) ∩ X

⊆

∪

(var(ˆ σ

[t

])) ∩ X

= ∪

(var(ˆ σ

[t

]) ∩ X

) ⊆ ∪

(var(t

) ∩ X

) =

∪

var(t

) ∩ X

= var(t) ∩ X

.

(ii) If t ∈ X, then (σ

◦

σ

)(f ) = t. So the term for σ

◦

σ

uses only one variable. Assume that t = f (t

, . . . , t

) and ˆ σ

[t

] uses only one variable for all 1 ≤ i ≤ n. So (σ

◦

σ

)(f ) = S

(s, ˆ σ

[t

], . . . , ˆ σ

[t

]). If var(s) = {x

} for some x

∈ X

, then var(σ

◦

σ

)(f ) = var(ˆ σ

[t

]). If var(s) = {x

} where i > n, then var(σ

◦

σ

)(f ) = var(s).

We conclude this section by extending the results from [8] to the case of Hyp

(n) on properties of the composition operation with a lemma describ- ing the special role of the terms π[t] and C

from Definition 4.2.

Lemma 4.7. For t ∈ W

(X).

(i) Let π be any permutation on J. Then σ

◦

σ

= σ

.

(ii) Let α be any function on J. Define the generalized hypersubstitu- tion σ

by mapping the fundemental f to the term f (x

, . . . , x

).

Then σ

◦

σ

= σ

.

P roof. (i) We will proceed by induction on the complexity of the term t. If t ∈ X, then by Lemma 2.1(i), σ

◦

σ

= σ

= σ

. Assume that t = f (t

, . . . , t

) and ˆ σ

[t

] = π[t

] for all 1 ≤ i ≤ n. So (σ

◦

σ

)(f ) = S

(f (x

, . . . , x

), ˆ σ

[t

], . . . , ˆ σ

[t

]) = f (ˆ σ

[t

], . . . , ˆ σ

[t

])

= f (π[t

], · · · , π[t

]) = π[f (t

, · · · , t

)] = π[t].

(ii) We have (σ

◦

σ

)(f ) = S

(t, x

, . . . , x

) = C

[t]. So σ

◦

σ

=

σ

.

5. Green’s ralations on Hyp G (n)

Let S be a semigroup and 1 / ∈ S. We extend the binary operation from S to S ∪ {1} by define x1 = 1x = x for all x ∈ S ∪ {1}. Then S ∪ {1} is a semigroup with identity 1.

Let S be a semigroup. Then we define,

S

=







S if S has an identity, S ∪ {1} otherwise.

Let S be a semigroup and ∅ 6= A ⊆ S. We now set (A)

= ∩{L|L is a left ideal of S containing A}

(A)

= ∩{R|R is a right ideal of S containing A}

(A)

= ∩{I|I is an ideal of S containing A}.

Then (A)

,(A)

and (A)

are left ideal, right ideal and ideal of S, respectively.

And we call (A)

((A)

, (A)

) the left ideal (right ideal, ideal) of S generated by A.

It is easy to see that

(A)

= S

A = SA ∪ A (A)

= AS

= A ∪ SA

(A)

= S

AS

= SAS ∪ SA ∪ AS ∪ A.

For a

, a

, . . . , a

∈ S, we write (a

, a

, . . . , a

)

instead of ({a

, a

, . . . , a

})

and call it the left ideal of S generated by a

, a

, . . . , a

. Similarly, we can define (a

, a

, . . . , a

)

and (a

, a

, . . . , a

)

. If A is a left ideal of S and A = (a)

for some a ∈ S, we then call A the principal left ideal generated by a. We can define principal right ideal and principal ideal in the same manner.

Let S be a semigroup. We define the relations L, R, H, D and J on S

as follow:

aLb ⇔ (a)

= (b)

aRb ⇔ (a)

= (b)

H = L ∩ R

D = L ◦ R aJ b ⇔ (a)

= (b)

. Then we have, for all a, b ∈ S

aLb ⇔ Sa ∪ {a} = Sb ∪ {b}

⇔ S

a = S

b

⇔ a = xb and b = ya for some x, y ∈ S

aRb ⇔ aS ∪ {a} = bS ∪ {b}

⇔ aS

= bS

⇔ a = bx and b = ay for some x, y ∈ S

aHb ⇔ aLb and aRb

aDb ⇔ (a, c) ∈ L and (c, b) ∈ R for some c ∈ S aJ b ⇔ SaS ∪ Sa ∪ aS ∪ {a} = SbS ∪ Sb ∪ bS ∪ {b}

⇔ S

aS

= S

bS

⇔ a = xby and b = zau for some x, y, z, u ∈ S

.

Remark 5.1. Let S be a semigroup. Then the following statements hold.

1. L, R, H, D and J are equivalent relations.

2. H ⊆ L ⊆ D ⊆ J and H ⊆ R ⊆ D ⊆ J .

We call the relations L, R, H, D and J the Green’s relations on S. For each a ∈ S, we denote L-class, R-class, H-class, D-class and J -class containing a by L

, R

, H

, D

and J

, respectively.

Green’s relations on Hyp(n) have been studied by S.L. Wismath [8], and Green’s relations on Hyp

(2) were study by W. Puninagool and S.

Leeratanavalee [5]. In this section, we describe some classes of the monoid of generalized hypersubstitutions of type τ = (n) with n > 1.

Theorem 5.2. Any σ

∈ P

(n) is L-related only to itself, but is R-related, D-related and J -related to all elements of P

(n), and not related to any other generalized hypersubstitutions. Moerover, the set P

(n) forms a com- plete R-, D- and J -class.

P roof. By Lemma 2.1(i), for any σ

∈ P

(n), σ ◦

σ

= σ

for all σ ∈ Hyp

(n). This shows that any σ

∈ P

(n) can be L-related only to itself. Since σ

◦

σ

= σ

for all σ

, σ

∈ P

(n), so any two elements in P

(n) are R-related. From R ⊆ D ⊆ J , we see that any two elements in P

(n) are D- and J -related. Moreover by Lemma 2.1, σ

◦

σ

◦

σ

∈ P

(n) for all σ

, σ

∈ Hyp

(n), and σ

∈ P

(n). This implies if σ / ∈ P

(n), then σ cannot be J -related to every element in P

(n). So P

(n) is the J -class of its elements. Since any two elements in P

(n) are R- and D-related, R ⊆ J , D ⊆ J and P

(n) is the J -class of its elements, and thus P

(n) forms a complete R-, D-class.

Theorem 5.3. Any σ

∈ G(n) is R-related only to itself, but is L-related, D-related and J -related to all elements of G(n), and not related to any other generalized hypersubstitutions. Moreover, the set G(n) forms a complete L-, D- and J -class.

P roof. Let σ

∈ G(n). Assume that σ

∈ Hyp

(n) where σ

Rσ

. By Theorem 5.2, s / ∈ X. Then there exists σ

∈ Hyp

(n) such that σ

= σ

◦

σ

. Since s / ∈ X and σ

= σ

◦

σ

, thus by Lemma 2.1(ii), p / ∈ X.

Since σ

∈ G(n) and p / ∈ X, thus by Lemma 3.1, σ

◦

σ

= σ

. So σ

= σ

. Thus σ

is R-related only to itself.

Let σ

, σ

∈ G(n). By Lemma 3.1, σ

◦

σ

= σ

and σ

◦

σ

= σ

. Thus

σ

Lσ

. So any two elements in G(n) are L-related. Since L ⊆ D ⊆ J , thus

any two elements in G(n) are D- and J -related. Assume that σ

∈ G(n)

and σ

∈ Hyp

(n) where σ

J σ

. By Theorem 5.2, s / ∈ X. Then there

exist σ

, σ

∈ Hyp

(n) such that σ

◦

σ

◦

σ

= σ

. Since s / ∈ X and σ

◦

σ

◦

σ

= σ

, thus by Lemma 2.1, p, q / ∈ X. Since σ

∈ G(n) and q / ∈ X, thus by Lemma 3.1, σ

◦

σ

= σ

. Since x

, . . . , x

∈ var(t), / thus by Lemma 4.6 (i), x

, . . . , x

are not variables occurring in the term (σ

◦

σ

)(f ) = (σ

◦

σ

◦

σ

)(f ). Thus x

, . . . , x

∈ var(s) and so σ /

∈ G(n). So G(n) is the J -class of its elements. Since any two elements in G(n) are L- and D- related, L ⊆ J , D ⊆ J and G(n) is the J -class of its elements, and thus G(n) forms a complete L-, D-class.

Lemma 5.4. Let σ

, σ

∈ Hyp

(n) \ (P

(n) ∪ G(n)). Then σ

Rσ

if and only if s = C

for some bijection α on J.

P roof. Assume that s = C

for some bijection α on J. So σ

and σ

are inverse generalized hypersubstitutions. So by Lemma 4.7(ii), σ

◦

σ

= σ

= σ

and σ

◦

σ

= σ

. Thus σ

Rσ

. Conversely, assume that σ

Rσ

. Then there exist p, q ∈ W

(X) \ X such that σ

◦

σ

= σ

and σ

◦

σ

= σ

. Let p = f (p

, . . . , p

) and q = f (q

, . . . , q

). So we have two equations

(1) S

(s, ˆ σ

[p

], . . . , ˆ σ

[p

]) = t

(2) S

(t, ˆ σ

[q

], . . . , ˆ σ

[q

]) = s.

Now, if neither of these equations satisfies the condition (Q) of Lemma 4.3, we would have the length of the term t is longer than the length of the term s and also the length of s is longer than the length of t, which is clearly impossible. Thus, at least one of two equations must fit the condition (Q).

But if one equation fits the condition (Q), Lemma 4.3 tells us that s and t have the same length, and therefore, the second equation also fits the condition (Q). By Lemma 4.3, if x

∈ var(t) ∩ X

, then q

∈ X. If such q

∈ X /

, then from (2) we get q

∈ var(s). So S

(s, ˆ σ

[p

], . . . , ˆ σ

[p

]) 6= t which contradicts to (1). Thus such q

∈ X

. Let α(i) = j if x

∈ var(t)∩X

and q

= x

. This defines a partial function on J. It is clear that α is injective. Extending this map to a bijection on J, which we shall also call α. So s = C

.

Lemma 5.5. Let t ∈ W

(X) and π be a permutation on J. Then π

[π[t]] = t.

P roof. We will proceed by induction on the complexity of the term t. If t ∈ X then π

[π[t]] = π

[t] = t. Assume that t = f (t

, . . . , t

) and π

[π[t

]] = t

for all 1 ≤ i ≤ n. So π

[π[t]] = π

[π[f (t

, . . . , t

)]] = π

[f (π[t

], . . . , π[t

])] = f (π

[π[t

]], . . . , π

[π[t

]]) = f (π

[π[t

]], . . . , π

[π[t

]]) = f (t

, . . . , t

) = t.

Lemma 5.6. Let σ

∈ Hyp

(n) \ P

(n). Then, for any permutation π on J, σ

is L-related to the generalized hypersubstitution σ

.

P roof. We know from Lemma 4.7(i) that σ

◦

σ

= σ

. From Lemma 4.7(i) and Lemma 5.4, σ

◦

σ

= σ

= σ

. So σ

Lσ

.

Lemma 5.7. Two idempotents σ

and σ

in Hyp

(n) \ P

(n) are L-related if and only if var(s) ∩ X

= var(t) ∩ X

.

P roof. Assume that σ

Lσ

. Then there exist u, v ∈ W

(X) such that σ

◦

σ

= σ

and σ

◦

σ

= σ

. By Lemma 4.6(i), var(s)∩X

⊆ var(t)∩X

and var(t) ∩ X

⊆ var(s) ∩ X

. So var(s) ∩ X

= var(t) ∩ X

. Conversely, we use the fact that for any two idempotents e and f in any semigroup, eLf if and only if ef = e and f e = f . Since var(s) ∩ X

= var(t) ∩ X

, by Theorem 3.2 we can prove that σ

◦

σ

= σ

and σ

◦

σ

= σ

.

Theorem 5.8. Let σ

be an idempotent in Hyp

(n) \ (P

(n) ∪ G(n)). Then L

= {σ

|π is a permutation of J, w / ∈ X, var(w) ∩ X

= var(t) ∩ X

and σ

is an idempotent}.

P roof. Let σ

∈ Hyp

(n) where π is a permutation of J, w / ∈ X, var(w)∩

X

= var(t) ∩ X

and σ

is an idempotent. By Lemma 5.7, σ

Lσ

. By Lemma 5.6, σ

Lσ

. So σ

Lσ

. Let t = f (u

, . . . , u

) and s = f (v

, . . . , v

) with σ

Lσ

. Then there exists f (b

, . . . , b

) ∈ W

(X) such that σ

◦

σ

= σ

. We write σ = σ

. From σ

◦

σ

= σ

, we get S

(f (b

, . . . , b

), ˆ

σ[v

], . . . , ˆ σ[v

]) = f (u

, . . . , u

). If x

∈ var(t) ∩ X

, then u

= x

since

σ

is an idempotent. So b

= x

for some x

∈ X

. This implies ˆ σ[v

] = x

and then v

= x

. Let β be a partial function on J defined by β(i) = j if

x

∈ var(t) ∩ X

and v

= x

. If β(i) = β(k) = j, then v

= x

= x

. So

i = k and β is injective. So β can be extended to a permutation α on J.

Let w = f (p

, . . . , p

) where p

= x

if x

∈ var(t) ∩ X

and p

= α[v

] if x

∈ var(t) ∩ X /

. We will show that var(w) ∩ X

= var(t) ∩ X

, σ

is an idempotent and s = f (v

, . . . , v

) = π[w] where w = α

. We show first that var(w) ∩ X

= var(t) ∩ X

. Since σ

Lσ

, thus by Lemma 5.7, var(s)∩X

= var(t)∩X

. Let x

∈ var(w)∩X

. Then x

∈ var(p

) for some i ∈ J and x

∈ X

. If p

= x

where x

∈ var(t) ∩ X

, then x

= x

∈ var(t).

If p

= α[v

], then x

∈ var(p

) = var(α[v

]) = var(v

) ⊆ var(s).

But var(s) ∩ X

= var(t) ∩ X

, so x

∈ var(t). Let x

∈ var(t) ∩ X

. Then p

= x

and so x

∈ var(s) ∩ X

. Next we show that σ

is an idempotent.

Let x

∈ var(w) ∩ X

. Then x

∈ var(t) ∩ X

. So p

= x

. Thus σ

is an idempotent. Finally we show that s = f (v

, . . . , v

) = π[w] where π = α

. To do this we will show that for all 1 ≤ k ≤ n, v

= π[p

]. Let 1 ≤ k ≤ n.

If there exists i ∈ J such that β(i) = k, then α(i) = k and π(k) = α

(k) = i. So p

= x

= v

. Thus π[p

] = π[p

] = π[x

] = x

= v

. If no such index i exists, then π[p

] = π[α[v

]] = π[α[v

]] = π[α[v

]] = α

[α[v

]] = v

.

Corollary 5.9. Let σ

be an idempotent in Hyp

(n)\(P

(n)∪G(n)). Then D

= {σ

|w = C

for some α bijection on J, π a permutation on J, s / ∈ X, and σ

an idempotent with var(s) ∩ X

= var(t) ∩ X

}.

Theorem 5.10. Let σ

be an idempotent in Hyp

(n)\(P

(n)∪G(n)). Then its J -class is equal to its D-class.

P roof. Let t = f (u

, . . . , u

) and let c be the number of distinct variables in X

which occur in t. Let s = f (v

, . . . , v

) with σ

J σ

. Then there exist f (a

, . . . , a

), f (b

, . . . , b

), f (p

, . . . , p

), f (r

, . . . , r

) ∈ W

(X) such that (1) σ

◦

σ

◦

σ

= σ

(2) σ

◦

σ

◦

σ

= σ

.

Let f (q

, . . . , q

) be the term for σ

◦

σ

. We write σ =

σ

. From (1), we get S

(f (a

, . . . , a

), ˆ σ[q

], . . . , ˆ σ[q

])=f (u

, . . . , u

).

If x

∈ var(t) ∩ X

, then u

= x

since σ

is an idempotent. So a

= x

for some x

∈ X

. This implies ˆ σ[q

] = x

and then q

= x

. Let α be a function from J(t) to J defined by α(k) = j if x

∈ var(t) ∩ X

and a

= x

where J(t) = {k ∈ J|x

∈ var(t)}. So α can be extended to a permutation on J.

We write σ

= σ

. Since f (q

, . . . , q

) is the term for σ

◦

σ

, thus S

(f (v

, . . . , v

), ˆ σ

[b

], . . . , ˆ σ

[b

]) = f (q

, . . . , q

). For each k ∈ J(t), q

= x

. So v

= x

for some x

∈ X

. So ˆ σ

[b

] = x

and then b

= x

. Let β : α(J(t)) → J defined by β(α(k)) = l where k ∈ J(t) and v

= x

. So β can be extended to a permutation on J.

Since α and β are injective, thus at least c distinct variables in X

occur as v

in entries of s = f (v

. . . , v

). We claim that the only variables in X

which occur in s are those c variables. Let f (c

, . . . , c

) be the term for σ

◦

σ

.

We write σ

= σ

. From (2), we get S

(f (p

, . . . , p

), ˆ σ

[c

], . . . , ˆ

σ

[c

]) = f (v

, . . . , v

). Since at least c distinct variables in X

occur as v

in entries s = f (v

. . . , v

), thus at least c distinct variables in X

occur as p

in entries s = f (p

. . . , p

) and then at least c distinct variables in X

occur as c

in entries f (c

. . . , c

).

We write σ

= σ

. Since f (c

, . . . , c

) is the term for σ

◦

σ

, thus S

(f (u

, . . . , u

), ˆ σ

[r

], . . . , ˆ σ

[r

]) = f (c

, . . . , c

). But f (u

, . . . , u

) has only c distinct variables in X

. Thus all the r

s used in the composition in (2) are variables in X

. So the number of distinct variables in X

which occur in f (v

, . . . , v

) is at most c. Thus the number of distinct variables in X

which occur in f (v

, . . . , v

) is c and every variable in X

which occurs in it occurs as a v

. Let w

= C

[f (v

, . . . , v

)].

So var(w

) ∩ X

= var(t) ∩ X

. From Lemma 5.4, we get σ

Rσ

. Let w

= α[w

]. From Lemma 5.4, σ

Lσ

. We will show that σ

is an idempotent. Let w

= C

[f (v

, . . . , v

)] = f (d

, . . . , d

). For each x

∈ var(t) ∩ X

, v

= x

. So d

= x

. From w

= α[w

], we get w

= α[f (d

, . . . , d

)] = f (α[d

], . . . , α[d

]) and var(w

) ∩ X

= var(t) ∩ X

. Let x

∈ var(w

) ∩ X

. Then x

∈ var(t) ∩ X

. So α[d

] = α[x

] = x

. So σ

is an idempotent. By Lemma 5.7, σ

Lσ

. So σ

Lσ

. Thus σ

Dσ

.

Corollary 5.11. Let σ

, σ

be idempotents in Hyp

(n) \ (P

(n) ∪ G(n)).

Then σ

and σ

are J - or D-related if and only if the number of distinct

variables in X

which occur in s and t are equal.

P roof. One direction follows immediately from Corollary 5.9. Conversely, let s = f (u

, . . . , u

), t = f (v

, . . . , v

), var(s) ∩ X

= {x

, . . . , x

} and var(t) ∩ X

= {x

, . . . , x

}. Since σ

, σ

are idempotents, thus u

= x

and v

= x

for all 1 ≤ j ≤ c. Let s

= f (u

, . . . , u

), t

= f (v

, . . . , v

) where u

j

= x

and v

j

= x

for all 1 ≤ j ≤ c and other u

= x

, v

= x

. By Lemma 5.7, σ

Lσ

and σ

Lσ

. Let π(l

) = k

for all 1 ≤ j ≤ c. Then π is injective. So π can be extended to a permutation on J, which we will also call π. So π[s

] = f (p

, . . . , p

) where p

= x

for all 1 ≤ j ≤ c and other p

= x

. Let α(k

) = l

for all 1 ≤ j ≤ c. So α can be extended to a bijection on J, which we will also call α. So C

= t

. Thus σ

J σ

J σ

J σ

.

Acknowledgements

This research was supported by the Graduate School and the Faculty of Science, Chiang Mai University, Thailand.

References

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[2] S. Leeratanavalee and K. Denecke, Generalized Hypersubstitutions and Strongly Solid Varieties, p. 135–145 in: General Algebra and Applications, Proc. of the “59 th Workshop on General Algebra”, ‘15 th Conference for Young Algebraists Potsdam 2000’, Shaker Verlag 2000.

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[4] W. Puninagool and S. Leeratanavalee, All Regular Elements in Hyp

(2), preprint 2009.

[5] W. Puninagool and S. Leeratanavalee, Green’s Relations on Hyp

(2), preprint 2009.

[6] W. Puninagool and S. Leeratanavalee, The Order of Generalized Hyper-

substitutions of Type τ = (2), International Journal of Mathematics and

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doi:10.1155/2008/263541.

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[8] S.L. Wismath, The monoid of hypersubstitutions of type (n), Southeast Asian Bull. Math. 24 (1) (2000), 115–128.

Received 4 November 2009

Revised 17 November 2009