# Green's-Like Relations on Algebras and Varieties

Authors: K. Denecke; S. L. Wismath
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## 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 typeτ.
Some sort of  (super-)associativity is needed for such definitions to work, and we consider algebras
which are clones of terms of type τ, where the clone axioms including superassociativity hold. This allows us to define
for any variety V of type τ two Green's-like
relations ℒV and ℛV on the term clone of type τ. We prove
a number of properties of these two relations, and describe their behaviour when
V is a variety of semigroups.


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## Body

## 1. Introduction

A semigroup is an algebra of type(2) 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 semigroup𝒜, we follow the convention of denoting the binary
operation of the semigroup by juxtaposition. For any elements a and b of A, we say that aℒAb if and only if  a=b or there exist
some c and d in A such that ca=b and db=a. When the semigroup 𝒜 is clear from
the context, we usually omit the superscript A on the name of
the relation ℒA and just write aℒb. Dual to this “left” relation is the
“right” relation ℛ defined by aℛb if and only if a=b or there exist c and d in A such that ac=b and bd=a. Both ℒ and ℛ are equivalence
relations on any semigroup 𝒜. The remaining Green's relations are ℋ=ℛ∩ℒ, 𝒟=ℛ o ℒ=ℒ o ℛ, and 𝒥, defined by a𝒥b if and only if a=b or there exist
elements c, d, p and q in A such that a=cbd and b=paq. 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
Section2, we propose some definitions for ℒ and ℛ, 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 ℒ and ℛ to relations ℒV and ℛV on the term
clone of any variety V. In Section 3, we deduce a number of properties of
these two relations, and then in Section 4 we examine their behaviour when V is a variety of
semigroups.


## 2. Green's Relations for Any Type

We begin with some notation. Throughout this paper, we
will assume a typeτ=(ni)i∈I, with an ni-ary operation
symbol fi for each index i in some set I. For each n≥1, we let Xn={x1,…,xn} be an n-element
alphabet of variables, and let Wτ(Xn) be the set of
all n-ary terms of
type τ. Then we set X={x1,x2,x3,…}, and let Wτ(X) denote the set
of all (finitary) terms of type τ. Terms can be represented by tree diagrams called
semantic trees. We will use the well-known Galois connection Id-Mod  between classes
of algebras and sets of identities. For any class K of algebras of
type τ and any set Σ of identities
of type τ, Mod Σ is the class of
all algebras 𝒜 of type τ which satisfy
all the identities in Σ, while IdK is the set of
all identities s≈t of type τ which are
satisfied by all algebras in K.As a preliminary step in defining Green's relations on
any algebra of arbitrary type, let us consider first the case of typeτ=(n), where we have a single operation symbol f of arity n≥1. In analogy with the two left and right Green's
relations ℒ and ℛ for type (2), we can define n different
Green's-like relations here. Let 𝒜 be an algebra
of type (n) and let a and b be elements of A. For each 1≤j≤n, set a𝒢jb if and only if a=b or there exist
elements b1,…,bj−1,bj+1,…,bn and a1,…,aj−1, aj+1, …,an in A such that(2.1)a=fA(b1,…,bj−1,b,bj+1,…,bn),b=fA(a1,…,aj−1,a,aj+1,…,an). Each 𝒢j for 1≤j≤n is clearly a
reflexive and symmetric relation on A, but as we will see is not necessarily transitive
for n≥2. Of particular interest are the two relations 𝒢1 and 𝒢n, which we will denote by ℛ and ℒ, respectively.Example 2.1.
Letτ=(1) be a type with
one unary operation symbol f. In this case ℒ=ℛ, and we see that for any algebra 𝒜=(A;fA) and any
elements a,b∈A, we  have aℒb if and only if a=b, or a=fA(b) and b=fA(a). Thus two distinct elements are related if and only if there is
a cycle between them in the
algebra 𝒜. The relation ℒ is transitive and hence an
equivalence relation: if aℒb and bℒc, and a≠b and b≠c, then we have a=fA(b), b=fA(a), b=fA(c), and c=fA(b). This forces a=c=fA(b), and so
aℒc. This also tells us that each element b∈A can be ℒ-related to at
most one element other than itself.
If the type(1) algebra 𝒜 has no cycles
in it, we get simply ℒ=ΔA, the diagonal relation on A. If A={a,b} with fA(a)=b and fA(b)=a, then ℒ=A×A. An algebra 𝒜 in which there
are some cycles but not every element that has a cycle will result in ℒ strictly
between ΔA and A×A.Now consider an algebra𝒜 of an arbitrary
type τ. Since there can be different operation symbols of
different arities in our type, we cannot define our relations 𝒢j using the jth 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.
Let𝒜
be any algebra
of type τ. We define relations ℛ and ℒ on A as follows. For
any a,b∈A, we set
(i)
aℛb if and only if a=b or a=fiA(b,b2,…,bni) and b=fkA(a,a2,…,ank),  for some i,k∈I and some
elements b2,…,bni and a2,…,ank in A.(ii)
aℒb if and only if a=b or a=fiA(b1,…,bni−1,b) and b=fkA(a1,…,ank−1,a),   for some i,k∈I and some
elements b1,…,bni−1 and a1,…,ank−1 in A.Again these two relations are clearly seen to be
reflexive and symmetric on the base setA of any algebra 𝒜. It is the requirement of transitivity that causes
problems, and forces us to impose some restrictions on our algebra. For
transitivity of ℛ on an algebra 𝒜, suppose that a, b, and c are in A,aℛb, and bℛc. In the special cases that a=b or b=c, we certainly have aℛc, so let us assume that a≠b and b≠c. Then we have a=fiA(b,b2,…,bni) and b=fkA(a,a2,…,ank), and also b=fpA(c,c2,…,cnp) and c=fqA(b,d1,…,dnq), for some operation symbols fi, fk, fp, and fq of our type and
some  elements b2,…,bni, a2,…,ank, c2,…cnp, d2,…,dnq of set A. By substitution, we get(2.2)a=fiA(fpA(c,c2,…,cnp),b2,…,bni),c=fqA(fkA(a,a2,…,ank),d1,…,dnq). But we need to be able to express a as fmA(c,e2,…,enm) for some
operation symbol fm and some
elements e2,…,enm. For type (2), this is dealt
with by the requirement that fA(fA(c,c2),b2) can be changed
to fA(c,fA(c2,b2)), that is, we have associativity in our algebra 𝒜. For arbitrary types, it would suffice here to have a
superassociative algebra, satisfying the superassociative law:(2.3)fi(fj(x1,x2,…,xnj),y2,…,yni)≈fj(x1,fi(x2,y2,…,yni),…,fi(xnj,y2,…,yni)).  This identity would allow us to express a as an element
with c in the
left-most position and similarly to express c in terms of a. Another way to handle this would be to define aℛb when a=b or a=t1A(b,b2,…,bn) and b=t2A(a,a2,…,am) for some term
operations t1A,t2A on 𝒜 and some
elements a2,…,am,b2,…,bn of A. In either approach, we are led to consider clones of
terms.Aclone 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 τ. This term clone is a heterogeneous or multi-based
algebra, having as universes or base sets the sets Wτ(Xn) of n-ary terms of
type τ, for n≥1. For each n≥1, the n variable terms x1,…,xn are selected as
nullary operations e1n,…,enn. And for each pair n,m of natural
numbers, there is a superposition operation Smn, from Wτ(Xn)×(Wτ(Xm))n to Wτ(Xm), defined by Smn(s,t1,…,tn)=s(t1,…,tn).This gives us the algebra(2.4)cloneτ:=(Wτ(Xn);Smn,ein)n,m≥1,1≤i≤n called the term clone of type τ. It satisfies the following three axioms called the clone axioms:(C1)
Smp(z,Smn(y1,x1,…,xn),…,Smn(yp,x1,…,xn))≈Smn(Snp(z,y1,…,yp),x1,…,xn), for m,n,p≥1;(C2)
Smn(ein(x1,…,xn))≈xi, for m,n≥1 and 1≤i≤n;(C3)
Snn(y,e1n,…,enn)≈y, for n≥1.Definition 2.3.
Letτ=(ni)i∈Ibe any type,
and let (Smn)n,m≥1be the
superposition operations on the term clone, cloneτ
.
One defines two relations ℛ and ℒ on cloneτ
as follows. For
any terms s and t in cloneτ, of arities
m and n, respectively,
(i)
sℛt if and only if s=t, or s=Snm(t,t1,…,tm) and t=Smn(s,s1,…,sn) for some terms t1,…,tm and s1,…,sn in cloneτ;(ii)
sℒt if and only if s=t, or m=n and s=Smm(t1,…,tm,t) and t=Smm(s1,…,sm,s) for some terms t1,…tm and s1,…sm in cloneτ.Lemma 2.4.
For any typeτ, the relation
ℛ
defined on cloneτ
is an
equivalence relation on
cloneτ.Proof.
As noted above, both relationsℛ and ℒ are reflexive
and symmetric by definition. Transitivity for ℛ follows from
the clone axiom (C1) as above.Transitivity ofℒ 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 relationℛ was defined by
Denecke and Jampachon in [3], but in the restricted special case of a Menger
algebra of rank n. These are algebras of type (n,0,…,0), having one n-ary operation
and n-nullary ones.
Menger algebras can be formed using terms as the following: the base set Wτ(Xn) of all n-ary terms of
type τ, along with the superposition operation Snn and the n-variable terms x1,…,xn, form a Menger algebra of rank n called the n-clone of type τ. Such algebras also satisfy the clone axioms (C1),
(C2), and (C3) (restricted to Snn). Denecke and
Jampachon also defined a left Green's-like relation as well, again on the
Menger algebra of rank n. Their left relation is a subset of our relation ℒ, and we will use the name ℒ¯ in the next
definition for the analogous relation in the term clone case.Now, we extend our definition of Green's relationsℒ and ℛ on cloneτ, to relations with respect to varieties of type τ.Definition 2.5.
LetV be any variety
of type τ. One defines relations ℛV, ℒV, and ℒV¯ on cloneτ
as follows. Let s and t be terms of
type τ, of arities m and n, respectively. Then
(i)
sℛVt if and only if s=t, or(2.5)s≈Snm(t,t1,…,tn)∈IdV,t≈Smn(s,s1,…sm)∈IdV for some terms t1,…,tn and s1,…,sm in cloneτ;(ii)
sℒVt if and only if n=m, and s=t or(2.6)s≈Smm(t1,t2,…,tm,t)∈IdV,t≈Smm(s1,s2,…sm,s)∈IdV for some terms t1,…,tm and s1,…,sm in cloneτ;(iii)
sℒV¯t if and only if n=m, and s=t or(2.7)s≈Smm(t1,t,…,t)∈IdV,t≈Smm(s1,s,…,s)∈IdV for some terms t1 and s1 in cloneτ.This definition actually includes Definition2.3 as a
special case: when V equals the
variety Alg(τ) of all algebras
of type τ, the relation IdV is simply
equality on cloneτ and we obtain
the relations of Definition 2.3. We remark that similar definitions could be
made for ℛ𝒜 and ℒ𝒜 for any algebra 𝒜, using identities of 𝒜. Another possible variation is to restrict the
existence of the terms t1,…,tn and s1,…,sm to terms from
some subclone C of cloneτ; in this case we could define subrelations ℛVC and ℒVC.The proof of the following lemma is similar to that of
Lemma2.4.Lemma 2.6.
For any typeτand any varietyVof typeτ, the relationℛVdefined oncloneτis an
equivalence relation oncloneτ.


## 3. The RelationsℛV and ℒV

In this section, we describe some properties of the
relationsℛV, ℒV, and ℒV¯, for any variety V. We begin with the relation ℒV.Proposition 3.1.
LetVbe any variety
of typeτ. Then
(i)
two terms of typeτ of arity, at
least two, are ℒV-related if and only if
they have the same arity;(ii)
the relationℒV is an
equivalence relation on the set Wτ(X) of all terms of
type τ.Proof.
(i)
It follows
from the definition of superposition of terms that the termSmn(t1,t2,…,tm,t) has the same
arity as t. Thus it is built into the definition of ℒV that any two
terms which are ℒV-related must
have the same arity. Conversely, let both s and t be terms of
arity n≥2. Then we can write s=Snn(x1,s,…,s,t) and t=Snn(x1,t,…,t,s), making s≈Snn(x1,s,…,s,t)∈IdV and t≈Snn(x1,t,…,t,s)∈IdV for any variety V, and so
sℒVt.
(ii) For any varietyV, ℒV 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 ℒV makes a
partition of Wτ(X) in which all
elements of Wτ(Xn) are related to
each other for n≥2. This means that it suffices to verify transitivity
for unary terms only. Let s, t, and u be unary terms
with sℒVt and tℒVu. Then there exist unary terms a, b, c, and d such that s≈S11(a,t), t≈S11(b,s), t≈S11(c,u), and u≈S11(d,t) all hold in IdV. Then by substitution and the clone axiom (C1), we
have s≈S11(a,S11(c,u))≈S11(S11(a,c),u) in IdV, and similarly u≈S11(S11(d,b),s) in IdV. This makes sℒVu as
required.We have shown that any two terms of the same arityn≥2 are ℒV-related, for
any variety V. Which unary terms are related, however, depends on
the variety V. For instance, if the operation fi is idempotent
in V, we can express the unary terms x1 and fi(x1,…,x1) in terms of
each other:(3.1)x1≈S11(x1,fi(x1,…,x1))∈IdV,fi(x1,…,x1)≈S11(f(x1,…,x1),x1)∈IdV. Thus x1 and fi(x1,…,x1) are ℒV-related when fi is idempotent;
but these terms need not be related if fi is not
idempotent. This question will be investigated in more detail in Section 4.Proposition 3.2.
LetAlg(τ)be the class of
all algebras of typeτ. The relationℒAlg(τ)¯is equal to the
identity relationΔWτ(X)onWτ(X).Proof.
This was
proved in [3] for the analogous relation ℒAlg(τ)¯ defined on the
rank n Menger algebra,
the n-clone of type τ. Since terms are ℒV¯-related only
if they have the same arity, the same proof covers the general term-clone case
as well.Example 3.3.
LetV be an
idempotent variety of type τ. Then it is easy to show that for any terms s and t of the same
arity n, we have Snn(s,t,…,t)≈t∈IdV. It follows from this that s≈Snn(p,t,…,t)∈IdV for some term p if and only if s≈t∈IdV. This means that for any terms s and t, we have sℒV¯t if and only if s and t have the same
arity and s≈t∈IdV. In particular, any two unary terms of type τ are ℒV¯-related in
this case. Combining this with Proposition 3.1 and the fact that ℒV¯⊆ℒV shows that when V is idempotent,
two terms are ℒV-related if and only if
they have the same arity. We see also that ℒV¯ is a proper
subset of ℒV when V is an
idempotent variety.Next we consider the right relationℛV. Denoting by ℒ(τ) the lattice of
all varieties of type τ, ordered by inclusion, we show first that ℛV is
order-reversing as an operator on ℒ(τ).Lemma 3.4.
(i)For any
varietiesU,W∈ℒ(τ), ifU⊆W, thenℛW⊆ℛU.
(ii)  IfℛV is equal to Wτ(X)2 for some
variety V, then ℛW=Wτ(X)2 for all
varieties W⊆V.Proof.
(i) Follows immediately from the fact thatIdW⊆IdU when U⊆W, and (ii) follows immediately from (i).Now we want to prove some facts about which pairs of
terms can beℛV-related.
Recall that X={x1,x2,x3,…} is the set of
all variables used in forming terms. Our first observation is that for any two
variables xj and xk of arities n and m, respectively, we can write xj=Smn(xk,xj,…,xj). This shows that any two variables are ℛV-related, for
any variety V; we write this as X×X⊆ℛV. Next suppose that s≈t is an identity
of V, with s of arity n and t of arity m. Then s≈Snm(t,x1,…,xm)∈IdV and t≈Smn(s,x1,…,xn)∈IdV, making sℛVt. Identifying the set IdV of all
identities of V with the subset {(s,t)∣s≈t∈IdV} of Wτ(X)2, we see that IdV⊆ℛV.Example 3.5.
LetV be the trivial
variety TRτ of type τ, defined by the identity x1≈x2. Then IdV=Wτ(X)2, since any identity is satisfied in V. From this and the previous comments, it follows that ℛV also equals Wτ(X)2 for this choice
of V.To further describeℛV, we need more notation. For any m≥1, let Symm be the
symmetric group of permutations of the set {1,2,…,m}. Let s=s(x1,…,xn) be an n-ary term. For
any m≥n and any
permutation π∈Symm, we will denote by π(s) the m-ary term Smn(s,xπ(1),…,xπ(n)). That is, π(s) is the term
formed from s by relabelling
the variables in s according to
the permutation π.Proposition 3.6.
letVbe any variety
of typeτ. For any termsof typeτof arityn, and any permutationπ∈Symm, wherem≥n, one getssℛVπ(s).Proof.
By definitionπ(s)=Smn(s,xπ(1),…,xπ(n)), so that π(s)≈Smn(s,xπ(1),…,xπ(n))∈IdV. For the other direction, to express s using π(s), we use the inverse permutation π−1∈Symm:(3.2)Snm(π(s),xπ−1(1),…,xπ−1(m))=Snm(Smn(s,xπ(1),…,xπ(n)),xπ−1(1),…,xπ−1(m))=Snn(s,Snm(xπ(1),xπ−1(1),…,xπ−1(m)),…,Snm(xπ(n),xπ−1(1),…,xπ−1(m)))by(C1)=Snn(s,x1,…,xn)=s.This gives an identity in IdV and shows that π(s)ℛVs.Definition 3.7.
LetΣ be any set of
identities. For any identity s≈t in Σ, with s of arity n and t of arity m, let π∈Symk and ρ∈Symr for k≥n and r≥m. Denote by Perm(Σ)
the set of all
pairs (π(s),ρ(t)) in Wτ(X)2 formed in this
way from identities s≈t in Σ.Proposition 3.8.
LetVbe any variety
of typeτ. Then(X×X)∪IdV⊆Perm(IdV)⊆ℛV.Proof.
First note that any identityxj≈xk in X×X can be produced
by applying two permutations π and ρ to the identity x1≈x1 from IdV, so we have X×X⊆Perm(IdV). The existence of identity permutations also gives us IdV⊆Perm(IdV).
Now lets≈t be an identity
of V, with π and ρ permutations on
the appropriate sets. We saw above that sℛVt, and by Proposition 3.6 also sℛVπ(s) and tℛVρ(t). By the symmetry and transitivity of ℛV we get π(s)ℛVρ(t). This shows that Perm(IdV)⊆ℛV.We note that as a consequence of Proposition3.8, the
equivalence relation ℛV is not in
general an equational theory on Wτ(X). The only equational theory in which any two
variables are related is IdV for V equal to the
trivial variety.Example 3.9.
In this
example we considerV=Alg(τ), the variety of all algebras of type τ. It is well-known that for this variety V, IdV=ΔWτ(X), the identity relation on Wτ(X); that is, an identity s≈t holds in V if and only if s=t. From Proposition, we know that Perm(ΔWτ(X)) is a subset of ℛV, and we will show that we have equality in this
case. Let s and t be terms of
arities n and m, respectively, and suppose that sℛVt. Without loss of generality, let us assume that n≥m. Then there exist terms t1,…,tm and s1,…,sn in Wτ(X) such that(3.3)s≈Snm(t,t1,…,tm)∈IdV,t≈Smn(s,s1,…sn)∈IdV. The property that IdV=ΔWτ(X) means that(3.4)s=Snm(t,t1,…,tm),t=Smn(s,s1,…sn). Then we have
(3.5)s=s(x1,…,xn)=Snm(t,t1,…,tm)=Snm(Smn(s,s1,…,sn),t1,…,tm)=Snn(s,Snm(s1,t1,…,tm),…,Snm(sn,t1,…,tm)),by(C1) This equality forces a strong condition on the entries
in the last line. Suppose that the variables occurring in term s are xi1,…,xik, with k≤n. Then we must have Snm(sij,t1,…,tm)=xij for each j=1,2,…,k. Then for each index ij there must
exist an index lj such that sij=xlj and tij=xij. Moreover the indices lj, for 1≤j≤k must be
distinct. This means that there is a permutation π on the set {1,2,…,n}, such that π(ij)=lj, for q≤j≤k. Then we have(3.6)t=t(x1,…,xm)=Smn(s,s1,…,sn)=Smn(s,s1,xl1,…,xl2,…,xlk,…,sn)=π(s), showing that we can obtain t by variable
permutation from s.Example 3.10.
A nontrivial varietyV of type τ is said to be
normal if it does not satisfy any identity of the form xj≈t, where xj is a variable
and t is a
nonvariable term. For each type τ, there is a smallest normal variety Nτ, which is defined by the set of identities {s≈t∣s,t∈Wτ(X)∖X}. That is, any two nonvariable terms are related by IdNτ, while each variable is related only to itself. Using
the fact that (X×X)∪IdV is always
contained in ℛV, we see that ℛNτ=(X×X)∪Wτ(X)2=Perm(IdNτ). This gives another
example of a variety V for which ℛV=Perm(IdV).We can use the relationℛV to characterize
when a variety V is normal.Proposition 3.11.
A varietyVof typeτis normal if and
only if no variable isℛV-related to a
nonvariable term.Proof.
WhenV is a normal
variety, we have Nτ⊆V and so by Lemma 3.4ℛV⊆ℛNτ. By the characterization of ℛNτ from Example
3.10 this means that no variable can be ℛV-related to a
nonvariable term. Conversely, suppose that ℛV has the
property that a variable can only be related to another variable. Since IdV⊆ℛV, this means that IdV cannot contain
any identity of the form xj≈t for xj a variable and t a nonvariable
term; in other words, V must be
normal.


## 4. The RelationℛV for Varieties of Semigroups

In this section we describe the relationsℛV and ℒV when V is a variety of
semigroups, that is, a variety of type (2) satisfying the
associative identity. We denote by Sem the variety Mod{x(yz)≈(xy)z} of all
semigroups. For any variety V, we use ℒ(V) for the lattice
of subvarieties of V; in particular ℒ(Sem) 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
“word” consisting of a string of variable symbols as letters; for
instance, the termf(x1,f(x2,f(x2,x1))) becomes the
word x1x2x2x1. 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 s≈t is called regular if the two terms s and t 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 V is called
regular if the set IdV of all its
identities is regular. A semigroup identity s≈t is called periodic if s=xa and t=xb for some
variable x and some
natural numbers a≠b. 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 s≈t have the
property that s and t have equal
lengths. For more information on uniformly periodic varieties, see [4].Lets=s(x1,…,xn) be a term of
some arity n≥1, and let π be a permutation
from Symm for some m≥n. In Section 3 we defined π(s) to be the term s(xπ(x1),…,xπ(xn)) formed from s by permutation
of the variables in s according to π. An important feature of this process is that the
term π(s) has the same
structure as the term s, in the sense that the semantic tree of the term π(s) is isomorphic
as a graph to the semantic tree for s. In particular, the term π(s) has the same
length and the same number of distinct variables occurring in it as s does. Which
variables occur need not be the same; for instance, s=x1x2 can be permuted
into π(s)=x3x4, changing the arity of the term and which variables
occur. As a result, a regular identity such as x1x2≈x2x1 can be permuted
by two different permutations π and ρ into a
nonregular identity such as x3x4≈x5x6. Thus the set Perm(IdV) from Section 3
need not be regular even when IdV is regular.
This motivates a new definition. We will call an identity s≈t permutation-regular if the number of distinct
variables occurring in s and t 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.
LetVbe a variety of
semigroups. IfVis regular,
thenPerm(IdV)is
permutation-regular.We saw in Section3 that any two terms of the same
arity n≥2 are ℒV-related, for
any variety V, and that only terms of the same arity can be ℒV-related. Thus
the only thing of interest for ℒV when V is a variety of
semigroups is which unary terms are related to each other. Let T1 denote the set
of unary semigroup terms, so that T1={xi|i≥1}.Proposition 4.2.
For any
varietyVof semigroups,ℒV∩T12=ℛV∩T12. That is, two unary terms areℒV-related if and
only if they areℛV-related.Proof.
Letxi and xj be two unary
terms, for i,j≥1, with i≠j. Then xiℛVxj
if and only if
xi≈S11(xj,xp) and xj≈S11(xi,xq) both hold in IdV, for some unary terms xp and xq. These identities hold if and only if xi≈xjp and xj≈xiq hold in V. Similarly, xiℒVxj if and only if xi≈S11(xp,xj) and xj≈S11(xq,xi) both hold in IdV, for some unary terms xp and xq, which is also equivalent to having both xi≈xjp and xj≈xiq in IdV.This result allows us to completely characterize the
relationℒV for V a variety of
semigroups, and begins our description of ℛV. Moreover, we have proved the following useful
characterization of when two unary terms are ℛV-related.Corollary 4.3.
LetVbe a variety of
semigroups and letxiandxjbe unary terms
withi≠j.ThenxiℛVxjif and only if the
identitiesxi≈xpjandxj≈xqihold inVfor some
natural numbersp,q≥1.Now we describe how the relationsℛV behave,
starting with unary terms.Proposition 4.4.
LetVbe a variety of
semigroups which is not uniformly periodic. ThenℒV∩T12=ℛV∩T12=ΔT1; that is, two unary terms are related byℛVif and only if they are
equal.Proof.
Letxi and xj be two unary
terms which are ℒV- or ℛV-related. By
Corollary 4.3, this forces identities of the form xi≈xpj and xj≈xqi to hold in V, for some natural numbers p and q. But when V is not
uniformly periodic, an identity of the form xa≈xb can hold in V if and only if a=b. Thus we must have i=pj and j=qi. This can only happen if i=j, and the terms xi and xj are in fact
equal.What happens with unary terms for uniformly periodic
varieties depends on the particular variety. We recall from Section3 that Perm(IdV)⊆ℛV. We will show that if V is both regular
and uniformly periodic, then IdV∩T12=Perm(IdV)∩T12, but ℛV∩T12 can be larger.Lemma 4.5.
IfV is a variety of
semigroups which is both regular and uniformly periodic, then IdV∩T12=Perm(IdV)∩T12.Proof.
SinceIdV⊆Perm(IdV) by definition,
we know that IdV∩T12⊆Perm(IdV)∩T12. For the opposite inclusion, suppose that xi≈xj is in Perm(IdV) for some unary
terms xi and xj. Then there exist some identity s≈t in IdV and some
permutations π and ρ such that xi=π(s) and xj=ρ(t). Since permutations do not change the number of
variables occurring or the length of a term, both s and t must look like xki and xmj, respectively, for some variables xk and xm. Since V is regular and s≈t is in IdV, the variables xk and xm must in fact be
the same. Therefore xi≈xj is actually in IdV.Any uniformly periodic varietyV must satisfy an
identity of the form xa≈xa+b for some
natural numbers a and b. We denote by Ba,b the variety Mod{x(yz)≈(xy)z,xa≈xa+b}, known as a Burnside variety. Thus any uniformly
periodic variety of semigroups is a subvariety of Ba,b for some a,b≥1. An important fact about the identities of the
variety Ba,b is the
following: an identity of the form xu≈xv holds in this
variety if and only if either u=v or both u,v≥a and u≡v modulo b. Combining this fact with Corollary 4.3 allows us to
describe which unary terms are ℛV-related for
the variety V=Ba,b.Corollary 4.6.
LetV=Ba,b, fora,b≥1. ThenxiℛVxjif and only if
bothi,j≥aand the
congruencesip≡jmodulobandjq≡imodulobhave solutionsp,q≥1.Some basic number theory now provides us with some
examples. Let us note that inV=Ba,b, the unary terms are (up to equivalence modulo IdV and hence
equivalence in ℛV as well) x,x2,…,xa+b−1. In the case a=b=1, we have all unary terms equivalent, and ℛV∩T12=T12. For V=Ba,1 or V=Ba,2, for any a≥1, it is easy to see that ℛV∩T12 is just IdV∩T12. But for V=B1,a when a is a prime
number, the terms x,x2,…,xa−1 are all ℛV-related to
each other, but not to xa; in this case more terms are related by ℛV than those
related by IdV. For V=B2.5, we can show that there are 3 distinct
classes of terms under ℛV: {x}, {x2,x3,x4,x6} and {x5}. This shows that for this choice of V, we have Perm(IdV)⊂ℛV⊂∇Wτ(X).Finally, we consider the relationℛV for terms of
arbitrary arity. Here too, uniformly periodic varieties behave differently from
those which are not uniformly periodic.Proposition 4.7.
IfVis a variety of
semigroups which is not uniformly periodic, thenℛV=Perm(IdV).Proof.
This proof is a modification of the argument from
Example3.9. First, by  Proposition 3.8 we have Perm(IdV)⊆ℛV, so we need to show the opposite inclusion. Let s and t be terms of
arities n and m, respectively, with n≥m, and suppose that sℛVt. Then there exist terms t1,…,tm and s1,…,sn in Wτ(X) such that(4.1)s≈Snm(t,t1,…,tm)∈IdV,t≈Smn(s,s1,…sn)∈IdV. Then we have(4.2)s≈Snm(t,t1,…,tm)≈Snm(Smn(s,s1,…,sn),t1,…,tm)≈Snn(s,Snm(s1,t1,…,tm),…,Snm(sn,t1,…,tm)),by(C1) Where in
Example 3.9 we have equality of terms, we now
have only equivalence modulo IdV. However, the condition that V is not
uniformly periodic means that the term Snn(s,Snm(s1,t1,…,tm),…,Snm(sn,t1,…,tm)) must have the
same length as s. This is sufficient to force the same requirement for
variable entries as before to produce our permutation π. Let the variables occurring in term s be xi1,…,xik, with k≤n. Then we must have Snm(sij,t1,…,tm)=xij for each j=1,2,…,k. Then for each index ij there must
exist an index lj such that sij=xlj and tij=xij. Moreover the indices lj, for 1≤j≤k must be distinct.
This means that there is a permutation π on the set {1,2,…,n}, such that π(ij)=lj, for q≤j≤k. Then we have(4.3)t=t(x1,…,xm)=Smn(s,s1,…,sn)=Smn(s,s1,xl1,…,xl2,…,xlk,…,sn)=π(s).This shows that t=π(s) for some
permutation π, and hence that ℛV⊆Perm(IdV).The converse of this proposition is not however true.
As an example we consider the smallest normal variety of type(2), the variety Zero of zero
semigroups defined by xy≈zw. This is a uniformly periodic but not regular
variety, but the relation ℛV for this
variety V is equal to Perm(IdV), from
Example 3.10.At the other extreme is the varietyB1,1 of idempotent
semigroups or bands. The lattice ℒ(B1,1) 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 ℛV is the total
relation ∇Wτ(X) on Wτ(X).Theorem 4.8.
LetVbe a variety of
semigroups. ThenℛV=∇Wτ(X)if and only ifVis a subvariety
of the varietyB1,1of bands.Proof.
First letV be a variety of
bands, so V⊆B1,1. Then it is easy to show by induction on the
complexity of terms that for any two terms s and t, of any arities n and m, respectively, we have s(t,t,…,t)≈t∈IdV. This means that we can always write t≈Smn(s,t,…,t)∈IdV and s≈Snm(t,s,…,s)∈IdV, making sℛVt.
Conversely, suppose thatV has the
property that any two terms (of any arities) are related by ℛV. Then the term x is related to
the term x2, so we must be able to express x≈S11(x2,p)∈IdV for some unary
term p=xc, for some c≥1. In particular, our variety V must satisfy an
identity of the form x≈xa for some a≥1. If a=1, we have x≈x2∈IdV, and we have shown that V is a variety of
bands. If a>1, then we can deduce the following identities from x≈xa :(4.4)x≈xa≈x2a−1≈x3a−2≈⋯∈IdVx2≈xa+1≈x2a≈x3a−1≈⋯∈IdVx3≈xa+2≈x2a+1≈x3a≈⋯∈IdV⋮⋮xa−1≈x2(a−1)≈x3(a−1)≈x4(a−1)≈⋯∈IdV. Now we also know that x is ℛV-related to xa−1, which means that we can write x≈S11(xa−1,q)∈IdV for some unary
term q=xk, for some k. Therefore, we get x≈xk(a−1)∈IdV. A similar argument applied to x2ℛVxa−1 then gives x2≈xm(a−1)∈IdV for some m. Since xk(a−1)≈xm(k−1) is in IdV from above, we
see that by transitivity we have x≈x2 in IdV, and V is a variety of
bands.Theorem 4.9.
LetV=Ba,bfor somea,b≥1. Lettbe any term of
arityn≥2which has at
least one variablexkoccurring in it
a number of times which is congruent to1modulob. ThenxaℛVta.Proof.
We can always writeta=Sn1(aa,p) for some n-ary term p, by taking p=t. But we also need to be able to write xa≈S1n(ta,q1,…,qn)∈IdV for some unary
terms q1,…,qn. Let xk be a variable
which occurs in t exactly v times, where v is congruent to 1 modulo b. For the term qk, we use x, and for all the other terms q1,…,qn, we use xb. Then S1n(ta,q1,…,qn)=ta(xb,…,xb,x,xb,…,xb)=(xqb+1)a for some
natural number q. Then in Ba,b we have (xqb+1)a≈xaqb+a≈xa, as required.Corollary 4.10.
LetV=Ba,bfor somea,b≥1witha+b≥3. ThenPerm(IdV)is a proper
subset ofℛV, which is a proper subset of∇Wτ(X)onWτ(X).Proof.
By the previous theorem, we havexaℛV(xy)a. Since the terms xa and (xy)a contain
different numbers of variables, and V is regular, the
identity xa≈(xy)a cannot be in Perm(IdV). Thus Perm(IdV) is a proper
subset of ℛV. The remaining claim follows from Theorem 4.8.



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