International Journal of Mathematics and Mathematical Sciences

Volumeย 2008ย (2008), Article IDย 362068, 12 pages

http://dx.doi.org/10.1155/2008/362068

## Green's-Like Relations on Algebras and Varieties

^{1}Institut fรผr Mathematik, Universitรคt Potsdam, D-14415 Potsdam, PF 601553, Germany^{2}Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB., Canada T1K 3M4

Received 13 September 2007; Accepted 30 October 2007

Academic Editor: Robert H.ย Redfield

Copyright ยฉ 2008 K. Denecke and S. L. Wismath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

There are five equivalence relations known as Green's relations definable on any semigroup or monoid, that is, on any algebra with a binary operation which is associative. In this paper, we examine whether Green's relations can be defined on algebras of any type . Some sort of (super-)associativity is needed for such definitions to work, and we consider algebras which are clones of terms of type , where the clone axioms including superassociativity hold. This allows us to define for any variety of type two Green's-like relations and on the term clone of type . We prove a number of properties of these two relations, and describe their behaviour when is a variety of semigroups.

#### 1. Introduction

A semigroup is an algebra of type for which the single binary operation satisfies the associativity identity. A monoid is a semigroup with an additional nullary operation which acts as an identity element for the binary operation. On any semigroup or monoid, the five equivalence relations known as Green's relations provide information about the structure of the semigroup.

To define Green's relations on a semigroup , we follow the convention of denoting the binary operation of the semigroup by juxtaposition. For any elements and of , we say that if and only if or there exist some and in such that and . When the semigroup is clear from the context, we usually omit the superscript on the name of the relation and just write . Dual to this โleftโ relation is the โrightโ relation defined by if and only if or there exist and in such that and . Both and are equivalence relations on any semigroup . The remaining Green's relations are , o o , and , defined by if and only if or there exist elements , , and in such that and . For more information about Green's relations in general, we refer the reader to [1].

In this paper, we consider how one might extend the definitions of the five Green's relations to algebras of any arbitrary type. In Section 2, we propose some definitions for and , and show what properties are needed to make our relations into equivalence relations. Then we consider a variation which extends our definition of two relations and to relations and on the term clone of any variety . In Section 3, we deduce a number of properties of these two relations, and then in Section 4 we examine their behaviour when is a variety of semigroups.

#### 2. Green's Relations for Any Type

We begin with some notation. Throughout this paper, we will assume a type , with an -ary operation symbol for each index in some set . For each we let be an -element alphabet of variables, and let be the set of all -ary terms of type . Then we set , and let denote the set of all (finitary) terms of type . Terms can be represented by tree diagrams called semantic trees. We will use the well-known Galois connection -Mod between classes of algebras and sets of identities. For any class of algebras of type and any set of identities of type , Mod is the class of all algebras of type which satisfy all the identities in , while is the set of all identities of type which are satisfied by all algebras in .

As a preliminary step in defining Green's relations on any algebra of arbitrary type, let us consider first the case of type , where we have a single operation symbol of arity . In analogy with the two left and right Green's relations and for type , we can define different Green's-like relations here. Let be an algebra of type and let and be elements of . For each , set if and only if or there exist elements and , , in such that Each for is clearly a reflexive and symmetric relation on , but as we will see is not necessarily transitive for . Of particular interest are the two relations and , which we will denote by and , respectively.

*Example 2.1. *Let be a type with
one unary operation symbol . In this case , and we see that for any algebra and any
elements , we have if and only if , or and . Thus two distinct elements are related if and only if there is
a cycle between them in the
algebra . The relation is transitive and hence an
equivalence relation: if and , and and , then we have , , and . This forces , and so
. This also tells us that each element can be -related to at
most one element other than itself.

If the type algebra has no cycles
in it, we get simply , the diagonal relation on . If with and , then . An algebra in which there
are some cycles but not every element that has a cycle will result in strictly
between and .

Now consider an algebra of an arbitrary type . Since there can be different operation symbols of different arities in our type, we cannot define our relations using the th position as before. But we can use the first and last position entries to define left and right relations. This motivates the following definition.

*Definition 2.2. *Let
be any algebra
of type . We define relations and on as follows. For
any , we set

(i) if and only if or and , for some and some
elements and in .(ii) if and only if or and , for some and some
elements and in .

Again these two relations are clearly seen to be reflexive and symmetric on the base set of any algebra . It is the requirement of transitivity that causes problems, and forces us to impose some restrictions on our algebra. For transitivity of on an algebra , suppose that , and are in and . In the special cases that or , we certainly have , so let us assume that and . Then we have and , and also and , for some operation symbols , , , and of our type and some elements , , , of set . By substitution, we get But we need to be able to express as for some operation symbol and some elements . For type this is dealt with by the requirement that can be changed to , that is, we have associativity in our algebra . For arbitrary types, it would suffice here to have a superassociative algebra, satisfying the superassociative law: This identity would allow us to express as an element with in the left-most position and similarly to express in terms of . Another way to handle this would be to define when or and for some term operations on and some elements of . In either approach, we are led to consider clones of terms.

A *clone* is an important kind of algebra which satisfies
a
superassociative law that we need here. Although
clones may be defined more generally (see [2]) we define here only the
*term clone* of type . This term clone is a heterogeneous or multi-based
algebra, having as universes or base sets the sets of -ary terms of
type , for . For each the variable terms are selected as
nullary operations . And for each pair of natural
numbers, there is a superposition operation , from to , defined by .

This gives us the algebra called the term clone of type . It satisfies the following three axioms called the *clone axioms*:

(C1), for (C2), for and ;(C3), for .

*Definition 2.3. *Let be any type,
and let be the
superposition operations on the term clone,
.
One defines two relations and on
as follows. For
any terms and in , of arities
and respectively,

(i) if and only if , or and for some terms and in ;(ii) if and only if , or and and for some terms and in .

Lemma 2.4. *
For any type
, the relation
defined on
is an
equivalence relation on
. *

*Proof. *As noted above, both relations and are reflexive
and symmetric by definition. Transitivity for follows from
the clone axiom (C1) as above.

Transitivity of does not follow directly from the clone axioms. We will show later that this relation is transitive, once we have deduced more information about it.

A similar definition of a Green's-like relation was defined by Denecke and Jampachon in [3], but in the restricted special case of a Menger algebra of rank . These are algebras of type , having one -ary operation and -nullary ones. Menger algebras can be formed using terms as the following: the base set of all -ary terms of type , along with the superposition operation and the -variable terms , form a Menger algebra of rank called the - of type . Such algebras also satisfy the clone axioms (C1), (C2), and (C3) (restricted to ). Denecke and Jampachon also defined a left Green's-like relation as well, again on the Menger algebra of rank . Their left relation is a subset of our relation , and we will use the name in the next definition for the analogous relation in the term clone case.

Now, we extend our definition of Green's relations and on , to relations with respect to varieties of type .

*Definition 2.5. *Let be any variety
of type . One defines relations , , and on
as follows. Let and be terms of
type , of arities and , respectively. Then

(i) if and only if , or for some terms and in ;(ii) if and only if , and or for some terms and in ;(iii) if and only if , and or for some terms and in .

This definition actually includes Definition 2.3 as a special case: when equals the variety of all algebras of type , the relation is simply equality on and we obtain the relations of Definition 2.3. We remark that similar definitions could be made for and for any algebra , using identities of . Another possible variation is to restrict the existence of the terms and to terms from some subclone of ; in this case we could define subrelations and .

The proof of the following lemma is similar to that of Lemma 2.4.

Lemma 2.6. *For any type ** and any variety ** of type **, the relation ** defined on ** is an
equivalence relation on **.
*

#### 3. The Relations and

In this section, we describe some properties of the relations , , and , for any variety . We begin with the relation .

Proposition 3.1. *Let ** be any variety
of type **. Then*

(i)*two terms of type of arity, at
least two, are -related if and only if
they have the same arity;*(ii)*the relation is an
equivalence relation on the set of all terms of
type .*

*Proof. *(i)
It follows
from the definition of superposition of terms that the term has the same
arity as . Thus it is built into the definition of that any two
terms which are -related must
have the same arity. Conversely, let both and be terms of
arity . Then we can write and , making and for any variety , and so
.

(ii) For any variety , is by
definition reflexive and symmetric, and we need only verify transitivity. Since
only elements of the same arity can be related, we see that makes a
partition of in which all
elements of are related to
each other for . This means that it suffices to verify transitivity
for unary terms only. Let , , and be unary terms
with and . Then there exist unary terms , , , and such that , , and all hold in . Then by substitution and the clone axiom (C1), we
have in , and similarly in . This makes as
required.

We have shown that any two terms of the same arity are -related, for any variety . Which unary terms are related, however, depends on the variety . For instance, if the operation is idempotent in , we can express the unary terms and in terms of each other: Thus and are -related when is idempotent; but these terms need not be related if is not idempotent. This question will be investigated in more detail in Section 4.

Proposition 3.2. *Let ** be the class of
all algebras of type **. The relation ** is equal to the
identity relation ** on *.

*Proof. *This was
proved in [3] for the analogous relation defined on the
rank Menger algebra,
the - of type . Since terms are -related only
if they have the same arity, the same proof covers the general term-clone case
as well.

*Example 3.3. *Let be an
idempotent variety of type . Then it is easy to show that for any terms and of the same
arity , we have . It follows from this that for some term if and only if . This means that for any terms and , we have if and only if and have the same
arity and . In particular, any two unary terms of type are -related in
this case. Combining this with Proposition 3.1 and the fact that shows that when is idempotent,
two terms are -related if and only if
they have the same arity. We see also that is a proper
subset of when is an
idempotent variety.

Next we consider the right relation . Denoting by the lattice of all varieties of type , ordered by inclusion, we show first that is order-reversing as an operator on .

Lemma 3.4. *
(i)
For any
varieties , if , then . *

*(ii) If is equal to for some variety , then for all varieties .*

*Proof. *(i) Follows immediately from the fact that when , and (ii) follows immediately from (i).

Now we want to prove some facts about which pairs of terms can be -related. Recall that is the set of all variables used in forming terms. Our first observation is that for any two variables and of arities and , respectively, we can write . This shows that any two variables are -related, for any variety ; we write this as . Next suppose that is an identity of , with of arity and of arity . Then and , making . Identifying the set of all identities of with the subset of , we see that .

*Example 3.5. *Let be the trivial
variety of type , defined by the identity . Then , since any identity is satisfied in . From this and the previous comments, it follows that also equals for this choice
of .

To further describe , we need more notation. For any , let be the symmetric group of permutations of the set . Let be an -ary term. For any and any permutation , we will denote by the -ary term . That is, is the term formed from by relabelling the variables in according to the permutation .

Proposition 3.6. *let ** be any variety
of type **. For any term** of type ** of arity **, and any permutation **, where **, one gets *.

*Proof. *By definition , so that . For the other direction, to express using , we use the inverse permutation :This gives an identity in and shows that .

*Definition 3.7. *Let be any set of
identities. For any identity in , with of arity and of arity , let and for and . Denote by
the set of all
pairs in formed in this
way from identities in .

Proposition 3.8. *Let ** be any variety
of type **. Then *.

*Proof. *First note that any identity in can be produced
by applying two permutations and to the identity from , so we have . The existence of identity permutations also gives us .

Now let be an identity
of , with and permutations on
the appropriate sets. We saw above that , and by Proposition 3.6 also and . By the symmetry and transitivity of we get . This shows that .

We note that as a consequence of Proposition 3.8, the equivalence relation is not in general an equational theory on . The only equational theory in which any two variables are related is for equal to the trivial variety.

*Example 3.9. *
In this
example we consider , the variety of all algebras of type . It is well-known that for this variety , , the identity relation on ; that is, an identity holds in if and only if . From Proposition, we know that is a subset of , and we will show that we have equality in this
case. Let and be terms of
arities and , respectively, and suppose that . Without loss of generality, let us assume that . Then there exist terms and in such that The property that means that Then we have

This equality forces a strong condition on the entries
in the last line. Suppose that the variables occurring in term are , with . Then we must have for each . Then for each index there must
exist an index such that and . Moreover the indices , for must be
distinct. This means that there is a permutation on the set , such that , for . Then we have showing that we can obtain by variable
permutation from .

*Example 3.10. * A nontrivial variety of type is said to be
normal if it does not satisfy any identity of the form , where is a variable
and is a
nonvariable term. For each type , there is a smallest normal variety , which is defined by the set of identities . That is, any two nonvariable terms are related by , while each variable is related only to itself. Using
the fact that is always
contained in , we see that . This gives another
example of a variety for which .

We can use the relation to characterize when a variety is normal.

Proposition 3.11. *A variety ** of type ** is normal if and
only if no variable is **-related to a
nonvariable term.*

*Proof. *When is a normal
variety, we have and so by Lemma 3.4 . By the characterization of from Example
3.10 this means that no variable can be -related to a
nonvariable term. Conversely, suppose that has the
property that a variable can only be related to another variable. Since , this means that cannot contain
any identity of the form for a variable and a nonvariable
term; in other words, must be
normal.

#### 4. The Relation for Varieties of Semigroups

In this section we describe the relations and when is a variety of
semigroups, that is, a variety of type satisfying the
associative identity. We denote by *Sem* the variety Mod of all
semigroups. For any variety , we use for the lattice
of subvarieties of ; in particular is the lattice
of all semigroup varieties.

We will follow the convention for semigroup varieties
of denoting the binary operation by juxtaposition, and of omitting brackets
from terms. In this way, any term can be represented by a semigroup
โwordโ consisting of a string of variable symbols as letters; for
instance, the term becomes the
word . We use this idea to define several properties of
terms and identities. The *length* of a term is its length as a word, the total number of occurrences of variables
in the term. An identity is called *regular* if the two terms and contain exactly
the same variable symbols. A set of identities is said to be regular if all the
identities in the set are regular, and a variety is called
regular if the set of all its
identities is regular. A semigroup identity is called *periodic* if and for some
variable and some
natural numbers . A variety of semigroups is called *uniformly periodic* if it satisfies a
periodic identity. A variety is not uniformly periodic if and only if all its
identities have the
property that and have equal
lengths. For more information on uniformly periodic varieties, see [4].

Let be a term of some arity , and let be a permutation from for some . In Section 3 we defined to be the term formed from by permutation of the variables in according to . An important feature of this process is that the term has the same structure as the term , in the sense that the semantic tree of the term is isomorphic as a graph to the semantic tree for . In particular, the term has the same length and the same number of distinct variables occurring in it as does. Which variables occur need not be the same; for instance, can be permuted into , changing the arity of the term and which variables occur. As a result, a regular identity such as can be permuted by two different permutations and into a nonregular identity such as . Thus the set from Section 3 need not be regular even when is regular. This motivates a new definition. We will call an identity permutation-regular if the number of distinct variables occurring in and is the same. As usual, a set of identities will be called permutation-regular if all the identities in the set are permutation-regular. We will make use of the following basic fact.

Lemma 4.1. *Let ** be a variety of
semigroups. If ** is regular,
then **is
permutation-regular.*

We saw in Section 3 that any two terms of the same arity are -related, for any variety , and that only terms of the same arity can be -related. Thus the only thing of interest for when is a variety of semigroups is which unary terms are related to each other. Let denote the set of unary semigroup terms, so that .

Proposition 4.2. *For any
variety ** of semigroups, **. That is, two unary terms are **-related if and
only if they are **-related.*

*Proof. *Let and be two unary
terms, for , with . Then
if and only if
and both hold in , for some unary terms and . These identities hold if and only if and hold in . Similarly, if and only if and both hold in , for some unary terms and , which is also equivalent to having both and in .

This result allows us to completely characterize the relation for a variety of semigroups, and begins our description of . Moreover, we have proved the following useful characterization of when two unary terms are -related.

Corollary 4.3. *Let ** be a variety of
semigroups and let ** and ** be unary terms
with **. ** Then ** if and only if the
identities ** and ** hold in ** for some
natural numbers **,
**.
*

Now we describe how the relations behave, starting with unary terms.

Proposition 4.4. *Let ** be a variety of
semigroups which is not uniformly periodic. Then **; that is, two unary terms are related by ** if and only if they are
equal.*

*Proof. *Let and be two unary
terms which are - or -related. By
Corollary 4.3, this forces identities of the form and to hold in , for some natural numbers and . But when is not
uniformly periodic, an identity of the form can hold in if and only if . Thus we must have and . This can only happen if , and the terms and are in fact
equal.

What happens with unary terms for uniformly periodic varieties depends on the particular variety. We recall from Section 3 that . We will show that if is both regular and uniformly periodic, then , but can be larger.

Lemma 4.5. *
If is a variety of
semigroups which is both regular and uniformly periodic, then . *

*Proof. *Since by definition,
we know that . For the opposite inclusion, suppose that is in for some unary
terms and . Then there exist some identity in and some
permutations and such that and . Since permutations do not change the number of
variables occurring or the length of a term, both and must look like and , respectively, for some variables and . Since is regular and is in the variables and must in fact be
the same. Therefore is actually in

Any uniformly periodic variety must satisfy an identity of the form for some natural numbers and . We denote by the variety Mod, known as a Burnside variety. Thus any uniformly periodic variety of semigroups is a subvariety of for some . An important fact about the identities of the variety is the following: an identity of the form holds in this variety if and only if either or both and modulo . Combining this fact with Corollary 4.3 allows us to describe which unary terms are -related for the variety .

Corollary 4.6. *Let **, for **. Then ** if and only if
both ** and the
congruences ** modulo ** and ** modulo ** have solutions *.

Some basic number theory now provides us with some examples. Let us note that in , the unary terms are (up to equivalence modulo and hence equivalence in as well) ,. In the case , we have all unary terms equivalent, and . For or , for any , it is easy to see that is just . But for when is a prime number, the terms , are all -related to each other, but not to ; in this case more terms are related by than those related by . For , we can show that there are distinct classes of terms under : , and . This shows that for this choice of , we have .

Finally, we consider the relation for terms of arbitrary arity. Here too, uniformly periodic varieties behave differently from those which are not uniformly periodic.

Proposition 4.7. *If ** is a variety of
semigroups which is not uniformly periodic, then *.

*Proof. *This proof is a modification of the argument from
Example 3.9. First, by Proposition 3.8 we have , so we need to show the opposite inclusion. Let and be terms of
arities and , respectively, with , and suppose that . Then there exist terms and in such that Then we have Where in
Example 3.9 we have equality of terms, we now
have only equivalence modulo . However, the condition that is not
uniformly periodic means that the term must have the
same length as . This is sufficient to force the same requirement for
variable entries as before to produce our permutation . Let the variables occurring in term be , with . Then we must have for each . Then for each index there must
exist an index such that and . Moreover the indices , for must be distinct.
This means that there is a permutation on the set , such that , for . Then we haveThis shows that for some
permutation , and hence that .

The converse of this proposition is not however true. As an example we consider the smallest normal variety of type , the variety of zero semigroups defined by . This is a uniformly periodic but not regular variety, but the relation for this variety is equal to , from Example 3.10.

At the other extreme is the variety of idempotent semigroups or bands. The lattice of band varieties is known to be countably infinite and its structure has been completely described by Birjukov [5], Fennemore [6, 7], Gerhard [8], and Gerhard and Petrich [9]. Our next result shows that varieties of bands are the only semigroup varieties for which is the total relation on .

Theorem 4.8. *Let ** be a variety of
semigroups. Then ** if and only if ** is a subvariety
of the variety ** of bands.*

*Proof. *First let be a variety of
bands, so . Then it is easy to show by induction on the
complexity of terms that for any two terms and , of any arities and , respectively, we have . This means that we can always write and , making .

Conversely, suppose that has the
property that any two terms (of any arities) are related by . Then the term is related to
the term , so we must be able to express for some unary
term , for some . In particular, our variety must satisfy an
identity of the form for some . If , we have , and we have shown that is a variety of
bands. If , then we can deduce the following identities from : Now we also know that is -related to , which means that we can write for some unary
term , for some . Therefore, we get . A similar argument applied to then gives for some . Since is in from above, we
see that by transitivity we have in , and is a variety of
bands.

Theorem 4.9. *Let ** for some **. Let ** be any term of
arity ** which has at
least one variable ** occurring in it
a number of times which is congruent to ** modulo **. Then *.

*Proof. * We can always write for some -ary term , by taking . But we also need to be able to write for some unary
terms . Let be a variable
which occurs in exactly times, where is congruent to modulo . For the term , we use , and for all the other terms , we use . Then for some
natural number . Then in we have , as required.

Corollary 4.10. *Let ** for some ** with **. Then ** is a proper
subset of **, which is a proper subset of **on*.

*Proof. * By the previous theorem, we have . Since the terms and contain
different numbers of variables, and is regular, the
identity cannot be in . Thus is a proper
subset of . The remaining claim follows from Theorem 4.8.

#### Acknowledgment

This research is supported by the NSERC of Canada.

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