Green's-Like Relations on Algebras and Varieties

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 𝜏. 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 ℒ_𝑉 and ℛ_𝑉 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.

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 semigroup 𝒜, 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 𝑎ℒ^𝐴𝑏 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 ℒ^𝐴 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 Section 2, 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 ℒ_𝑉 and ℛ_𝑉 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 type 𝜏=(𝑛_𝑖)_𝑖∈𝐼, with an 𝑛_𝑖-ary operation symbol 𝑓_𝑖 for each index 𝑖 in some set 𝐼. For each 𝑛≥1, we let 𝑋_𝑛={𝑥₁,…,𝑥_𝑛} be an 𝑛-element alphabet of variables, and let 𝑊_𝜏(𝑋_𝑛) be the set of all 𝑛-ary terms of type 𝜏. Then we set 𝑋={𝑥₁,𝑥₂,𝑥₃,…}, and let 𝑊_𝜏(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 type 𝜏=(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 𝑎𝒢_𝑗𝑏 if and only if 𝑎=𝑏 or there exist elements 𝑏₁,…,𝑏_𝑗−1,𝑏_𝑗+1,…,𝑏_𝑛 and 𝑎₁,…,𝑎_𝑗−1, 𝑎_𝑗+1, …,𝑎_𝑛 in 𝐴 such that𝑎=𝑓^𝐴𝑏¹,…,𝑏_𝑗−1,𝑏,𝑏_𝑗+1,…,𝑏_𝑛,𝑏=𝑓^𝐴𝑎¹,…,𝑎_𝑗−1,𝑎,𝑎_𝑗+1,…,𝑎_𝑛.(2.1) Each 𝒢_𝑗 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 𝒢_𝑛, which we will denote by ℛ and ℒ, respectively.

Example 2.1. Let 𝜏=(1) be a type with one unary operation symbol 𝑓. In this case ℒ=ℛ, and we see that for any algebra 𝒜=(𝐴;𝑓^𝐴) and any elements 𝑎,𝑏∈𝐴, we have 𝑎ℒ𝑏 if and only if 𝑎=𝑏, or 𝑎=𝑓^𝐴(𝑏) and 𝑏=𝑓^𝐴(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=𝑓^𝐴(𝑏), 𝑏=𝑓^𝐴(𝑎), 𝑏=𝑓^𝐴(𝑐), and 𝑐=𝑓^𝐴(𝑏). This forces 𝑎=𝑐=𝑓^𝐴(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=Δ_𝐴, the diagonal relation on 𝐴. If 𝐴={𝑎,𝑏} with 𝑓^𝐴(𝑎)=𝑏 and 𝑓^𝐴(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 Δ_𝐴 and 𝐴×𝐴.

Now consider an algebra 𝒜 of an arbitrary type 𝜏. Since there can be different operation symbols of different arities in our type, we cannot define our relations 𝒢_𝑗 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. Let 𝒜 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=𝑓^𝐴𝑖(𝑏,𝑏₂,…,𝑏_𝑛𝑖) and 𝑏=𝑓^𝐴𝑘(𝑎,𝑎₂,…,𝑎_𝑛𝑘), for some 𝑖,𝑘∈𝐼 and some elements 𝑏₂,…,𝑏_𝑛𝑖 and 𝑎₂,…,𝑎_𝑛𝑘 in 𝐴.(ii)𝑎ℒ𝑏 if and only if 𝑎=𝑏 or 𝑎=𝑓^𝐴𝑖(𝑏₁,…,𝑏_𝑛𝑖−1,𝑏) and 𝑏=𝑓^𝐴𝑘(𝑎₁,…,𝑎_𝑛𝑘−1,𝑎), for some 𝑖,𝑘∈𝐼 and some elements 𝑏₁,…,𝑏_𝑛𝑖−1 and 𝑎₁,…,𝑎_𝑛𝑘−1 in 𝐴.

Again these two relations are clearly seen to be reflexive and symmetric on the base set 𝐴 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=𝑓^𝐴𝑖(𝑏,𝑏₂,…,𝑏_𝑛𝑖) and 𝑏=𝑓^𝐴𝑘(𝑎,𝑎₂,…,𝑎_𝑛𝑘), and also 𝑏=𝑓^𝐴𝑝(𝑐,𝑐₂,…,𝑐_𝑛𝑝) and 𝑐=𝑓^𝐴𝑞(𝑏,𝑑₁,…,𝑑_𝑛𝑞), for some operation symbols 𝑓_𝑖, 𝑓_𝑘, 𝑓_𝑝, and 𝑓_𝑞 of our type and some elements 𝑏₂,…,𝑏_𝑛𝑖, 𝑎₂,…,𝑎_𝑛𝑘, 𝑐₂,…𝑐_𝑛𝑝, 𝑑₂,…,𝑑_𝑛𝑞 of set 𝐴. By substitution, we get𝑎=𝑓^𝐴𝑖𝑓^𝐴𝑝𝑐,𝑐²,…,𝑐_𝑛𝑝,𝑏²,…,𝑏_𝑛𝑖,𝑐=𝑓^𝐴𝑞𝑓^𝐴𝑘𝑎,𝑎²,…,𝑎_𝑛𝑘,𝑑¹,…,𝑑_𝑛𝑞.(2.2) But we need to be able to express 𝑎 as 𝑓^𝐴𝑚(𝑐,𝑒₂,…,𝑒_𝑛𝑚) for some operation symbol 𝑓_𝑚 and some elements 𝑒₂,…,𝑒_𝑛𝑚. For type (2), this is dealt with by the requirement that 𝑓^𝐴(𝑓^𝐴(𝑐,𝑐₂),𝑏₂) can be changed to 𝑓^𝐴(𝑐,𝑓^𝐴(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:𝑓_𝑖𝑓^𝑗𝑥¹,𝑥₂,…,𝑥_𝑛𝑗,𝑦²,…,𝑦_𝑛𝑖≈𝑓^𝑗𝑥¹,𝑓_𝑖𝑥²,𝑦₂,…,𝑦_𝑛𝑖,…,𝑓^𝑖𝑥^𝑛_𝑗,𝑦₂,…,𝑦_𝑛𝑖.(2.3) 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=𝑡^𝐴1(𝑏,𝑏₂,…,𝑏_𝑛) and 𝑏=𝑡^𝐴2(𝑎,𝑎₂,…,𝑎_𝑚) for some term operations 𝑡^𝐴1,𝑡^𝐴2 on 𝒜 and some elements 𝑎₂,…,𝑎_𝑚,𝑏₂,…,𝑏_𝑛 of 𝐴. 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 𝜏. This term clone is a heterogeneous or multi-based algebra, having as universes or base sets the sets 𝑊_𝜏(𝑋_𝑛) of 𝑛-ary terms of type 𝜏, for 𝑛≥1. For each 𝑛≥1, the 𝑛 variable terms 𝑥₁,…,𝑥_𝑛 are selected as nullary operations 𝑒^𝑛1,…,𝑒^𝑛𝑛. And for each pair 𝑛,𝑚 of natural numbers, there is a superposition operation 𝑆^𝑛𝑚, from 𝑊_𝜏(𝑋_𝑛)×(𝑊_𝜏(𝑋_𝑚))^𝑛 to 𝑊_𝜏(𝑋_𝑚), defined by 𝑆^𝑛𝑚(𝑠,𝑡₁,…,𝑡_𝑛)=𝑠(𝑡₁,…,𝑡_𝑛).

This gives us the algebraclone𝜏∶=𝑊_𝜏𝑋^𝑛;𝑆^𝑛𝑚,𝑒^𝑛𝑖_{𝑛,𝑚≥1,1≤𝑖≤𝑛}(2.4) called the term clone of type 𝜏. It satisfies the following three axioms called the clone axioms:

(C1)𝑆^𝑝𝑚(𝑧,𝑆^𝑛𝑚(𝑦₁,𝑥₁,…,𝑥_𝑛),…,𝑆^𝑛𝑚(𝑦_𝑝,𝑥₁,…,𝑥_𝑛))≈𝑆^𝑛𝑚(𝑆^𝑝𝑛(𝑧,𝑦₁,…,𝑦_𝑝),𝑥₁,…,𝑥_𝑛), for 𝑚,𝑛,𝑝≥1;(C2)𝑆^𝑛𝑚(𝑒^𝑛𝑖(𝑥₁,…,𝑥_𝑛))≈𝑥_𝑖, for 𝑚,𝑛≥1 and 1≤𝑖≤𝑛;(C3)𝑆^𝑛𝑛(𝑦,𝑒^𝑛1,…,𝑒^𝑛𝑛)≈𝑦, for 𝑛≥1.

Definition 2.3. Let 𝜏=(𝑛_𝑖)_𝑖∈𝐼be any type, and let (𝑆^𝑛𝑚)_{𝑛,𝑚≥1}be 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=𝑆^𝑚𝑛(𝑡,𝑡₁,…,𝑡_𝑚) and 𝑡=𝑆^𝑛𝑚(𝑠,𝑠₁,…,𝑠_𝑛) for some terms 𝑡₁,…,𝑡_𝑚 and 𝑠₁,…,𝑠_𝑛 in clone𝜏;(ii)𝑠ℒ𝑡 if and only if 𝑠=𝑡, or 𝑚=𝑛 and 𝑠=𝑆^𝑚𝑚(𝑡₁,…,𝑡_𝑚,𝑡) and 𝑡=𝑆^𝑚𝑚(𝑠₁,…,𝑠_𝑚,𝑠) for some terms 𝑡₁,…𝑡_𝑚 and 𝑠₁,…𝑠_𝑚 in clone𝜏.

Lemma 2.4. For any type 𝜏, the relation ℛ defined on clone𝜏 is an equivalence relation on clone𝜏.

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

Transitivity of ℒ 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 ℛ 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 𝑊_𝜏(𝑋_𝑛) of all 𝑛-ary terms of type 𝜏, along with the superposition operation 𝑆^𝑛𝑛 and the 𝑛-variable terms 𝑥₁,…,𝑥_𝑛, 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 𝑆^𝑛𝑛). 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 relations ℒ and ℛ on clone𝜏, to relations with respect to varieties of type 𝜏.

Definition 2.5. Let 𝑉 be any variety of type 𝜏. One defines relations ℛ_𝑉, ℒ_𝑉, and ℒ_𝑉 on clone𝜏 as follows. Let 𝑠 and 𝑡 be terms of type 𝜏, of arities 𝑚 and 𝑛, respectively. Then
(i)𝑠ℛ_𝑉𝑡 if and only if 𝑠=𝑡, or𝑠≈𝑆^𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑛∈Id𝑉,𝑡≈𝑆^𝑛𝑚𝑠,𝑠¹,…𝑠_𝑚∈Id𝑉(2.5) for some terms 𝑡₁,…,𝑡_𝑛 and 𝑠₁,…,𝑠_𝑚 in clone𝜏;(ii)𝑠ℒ_𝑉𝑡 if and only if 𝑛=𝑚, and 𝑠=𝑡 or𝑠≈𝑆^𝑚𝑚𝑡¹,𝑡₂,…,𝑡_𝑚,𝑡∈Id𝑉,𝑡≈𝑆^𝑚𝑚𝑠¹,𝑠₂,…𝑠_𝑚,𝑠∈Id𝑉(2.6) for some terms 𝑡₁,…,𝑡_𝑚 and 𝑠₁,…,𝑠_𝑚 in clone𝜏;(iii)𝑠ℒ_𝑉𝑡 if and only if 𝑛=𝑚, and 𝑠=𝑡 or𝑠≈𝑆^𝑚𝑚𝑡¹,𝑡,…,𝑡∈Id𝑉,𝑡≈𝑆^𝑚𝑚𝑠¹,𝑠,…,𝑠∈Id𝑉(2.7) for some terms 𝑡₁ and 𝑠₁ in clone𝜏.

This definition actually includes Definition 2.3 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 ℛ_𝒜 and ℒ_𝒜 for any algebra 𝒜, using identities of 𝒜. Another possible variation is to restrict the existence of the terms 𝑡₁,…,𝑡_𝑛 and 𝑠₁,…,𝑠_𝑚 to terms from some subclone 𝐶 of clone𝜏; in this case we could define subrelations ℛ^𝐶𝑉 and ℒ^𝐶𝑉.

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

Lemma 2.6. For any type 𝜏 and any variety 𝑉 of type 𝜏, the relation ℛ_𝑉 defined on clone𝜏 is an equivalence relation on clone𝜏.

3. The Relations ℛ_𝑉 and ℒ_𝑉

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

Proposition 3.1. Let 𝑉 be any variety of type 𝜏. Then
(i)two terms of type 𝜏 of arity, at least two, are ℒ_𝑉-related if and only if they have the same arity;(ii)the relation ℒ_𝑉 is an equivalence relation on the set 𝑊_𝜏(𝑋) of all terms of type 𝜏.

Proof. (i) It follows from the definition of superposition of terms that the term 𝑆^𝑛𝑚(𝑡₁,𝑡₂,…,𝑡_𝑚,𝑡) has the same arity as 𝑡. Thus it is built into the definition of ℒ_𝑉 that any two terms which are ℒ_𝑉-related must have the same arity. Conversely, let both 𝑠 and 𝑡 be terms of arity 𝑛≥2. Then we can write 𝑠=𝑆^𝑛𝑛(𝑥₁,𝑠,…,𝑠,𝑡) and 𝑡=𝑆^𝑛𝑛(𝑥₁,𝑡,…,𝑡,𝑠), making 𝑠≈𝑆^𝑛𝑛(𝑥₁,𝑠,…,𝑠,𝑡)∈Id𝑉 and 𝑡≈𝑆^𝑛𝑛(𝑥₁,𝑡,…,𝑡,𝑠)∈Id𝑉 for any variety 𝑉, and so 𝑠ℒ_𝑉𝑡.
(ii) For any variety 𝑉, ℒ_𝑉 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 ℒ_𝑉 makes a partition of 𝑊_𝜏(𝑋) in which all elements of 𝑊_𝜏(𝑋_𝑛) 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 𝑠ℒ_𝑉𝑡 and 𝑡ℒ_𝑉𝑢. 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 𝑠ℒ_𝑉𝑢 as required.

We have shown that any two terms of the same arity 𝑛≥2 are ℒ_𝑉-related, for any variety 𝑉. Which unary terms are related, however, depends on the variety 𝑉. For instance, if the operation 𝑓_𝑖 is idempotent in 𝑉, we can express the unary terms 𝑥₁ and 𝑓_𝑖(𝑥₁,…,𝑥₁) in terms of each other:𝑥₁≈𝑆¹¹𝑥¹,𝑓_𝑖𝑥¹,…,𝑥₁∈Id𝑉,𝑓^𝑖𝑥¹,…,𝑥₁≈𝑆¹¹𝑓𝑥¹,…,𝑥₁,𝑥¹∈Id𝑉.(3.1) Thus 𝑥₁ and 𝑓_𝑖(𝑥₁,…,𝑥₁) are ℒ_𝑉-related when 𝑓_𝑖 is idempotent; but these terms need not be related if 𝑓_𝑖 is not idempotent. This question will be investigated in more detail in Section 4.

Proposition 3.2. Let 𝐴𝑙𝑔(𝜏) be the class of all algebras of type 𝜏. The relation ℒ_{𝐴𝑙𝑔(𝜏)} is equal to the identity relation Δ_{𝑊𝜏(𝑋)} on 𝑊_𝜏(𝑋).

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

Example 3.3. Let 𝑉 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 𝑆^𝑛𝑛(𝑠,𝑡,…,𝑡)≈𝑡∈Id𝑉. It follows from this that 𝑠≈𝑆^𝑛𝑛(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 𝑠ℒ_𝑉𝑡 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 ℒ_𝑉-related in this case. Combining this with Proposition 3.1 and the fact that ℒ_𝑉⊆ℒ_𝑉 shows that when 𝑉 is idempotent, two terms are ℒ_𝑉-related if and only if they have the same arity. We see also that ℒ_𝑉 is a proper subset of ℒ_𝑉 when 𝑉 is an idempotent variety.

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

Lemma 3.4. (i) For any varieties 𝑈,𝑊∈ℒ(𝜏), if 𝑈⊆𝑊, then ℛ_𝑊⊆ℛ_𝑈.
(ii) If ℛ_𝑉 is equal to 𝑊_𝜏(𝑋)² for some variety 𝑉, then ℛ_𝑊=𝑊_𝜏(𝑋)² for all varieties 𝑊⊆𝑉.

Proof. (i) Follows immediately from the fact that Id𝑊⊆Id𝑈 when 𝑈⊆𝑊, and (ii) follows immediately from (i).

Now we want to prove some facts about which pairs of terms can be ℛ_𝑉-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 𝑥_𝑗 and 𝑥_𝑘 of arities 𝑛 and 𝑚, respectively, we can write 𝑥_𝑗=𝑆^𝑛𝑚(𝑥_𝑘,𝑥_𝑗,…,𝑥_𝑗). This shows that any two variables are ℛ_𝑉-related, for any variety 𝑉; we write this as 𝑋×𝑋⊆ℛ_𝑉. Next suppose that 𝑠≈𝑡 is an identity of 𝑉, with 𝑠 of arity 𝑛 and 𝑡 of arity 𝑚. Then 𝑠≈𝑆^𝑚𝑛(𝑡,𝑥₁,…,𝑥_𝑚)∈Id𝑉 and 𝑡≈𝑆^𝑛𝑚(𝑠,𝑥₁,…,𝑥_𝑛)∈Id𝑉, making 𝑠ℛ_𝑉𝑡. Identifying the set Id𝑉 of all identities of 𝑉 with the subset {(𝑠,𝑡)∣𝑠≈𝑡∈Id𝑉} of 𝑊_𝜏(𝑋)², we see that Id𝑉⊆ℛ_𝑉.

Example 3.5. Let 𝑉 be the trivial variety TR_𝜏 of type 𝜏, defined by the identity 𝑥₁≈𝑥₂. Then Id𝑉=𝑊_𝜏(𝑋)², since any identity is satisfied in 𝑉. From this and the previous comments, it follows that ℛ_𝑉 also equals 𝑊_𝜏(𝑋)² for this choice of 𝑉.

To further describe ℛ_𝑉, we need more notation. For any 𝑚≥1, let 𝑆𝑦𝑚_𝑚 be the symmetric group of permutations of the set {1,2,…,𝑚}. Let 𝑠=𝑠(𝑥₁,…,𝑥_𝑛) be an 𝑛-ary term. For any 𝑚≥𝑛 and any permutation 𝜋∈𝑆𝑦𝑚_𝑚, we will denote by 𝜋(𝑠) the 𝑚-ary term 𝑆^𝑛𝑚(𝑠,𝑥_𝜋(1),…,𝑥_𝜋(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. let 𝑉 be any variety of type 𝜏. For any term𝑠 of type 𝜏 of arity 𝑛, and any permutation 𝜋∈𝑆𝑦𝑚_𝑚, where 𝑚≥𝑛, one gets 𝑠ℛ_𝑉𝜋(𝑠).

Proof. By definition 𝜋(𝑠)=𝑆^𝑛𝑚(𝑠,𝑥_𝜋(1),…,𝑥_𝜋(𝑛)), so that 𝜋(𝑠)≈𝑆^𝑛𝑚(𝑠,𝑥_𝜋(1),…,𝑥_𝜋(𝑛))∈Id𝑉. For the other direction, to express 𝑠 using 𝜋(𝑠), we use the inverse permutation 𝜋⁻¹∈𝑆𝑦𝑚_𝑚:𝑆^𝑚𝑛=𝜋𝑆𝑠_𝑚𝑛,𝑥^𝜋−1(1),…,𝑥_{𝜋−1(𝑚)}=𝑆_𝑛𝑛𝑆^𝑛𝑚𝑠,𝑥^𝜋(1),…,𝑥_𝜋(𝑛),𝑥^𝜋−1(1),…,𝑥_{𝜋−1(𝑚)}=𝑆_𝑛𝑛𝑠,𝑆^𝑚𝑛𝑥^𝜋(1),𝑥_𝜋−1(1),…,𝑥_{𝜋−1(𝑚)},…,𝑆^𝑚𝑛𝑥^𝜋(𝑛),𝑥_𝜋−1(1),…,𝑥_{𝜋−1(𝑚)}byC1𝑠,𝑥¹,…,𝑥_𝑛=𝑠.(3.2)This gives an identity in Id𝑉 and shows that 𝜋(𝑠)ℛ_𝑉𝑠.

Definition 3.7. Let Σ be any set of identities. For any identity 𝑠≈𝑡 in Σ, with 𝑠 of arity 𝑛 and 𝑡 of arity 𝑚, let 𝜋∈𝑆𝑦𝑚_𝑘 and 𝜌∈𝑆𝑦𝑚_𝑟 for 𝑘≥𝑛 and 𝑟≥𝑚. Denote by 𝑃𝑒𝑟𝑚(Σ) the set of all pairs (𝜋(𝑠),𝜌(𝑡)) in 𝑊_𝜏(𝑋)² formed in this way from identities 𝑠≈𝑡 in Σ.

Proposition 3.8. Let 𝑉 be any variety of type 𝜏. Then (𝑋×𝑋)∪Id𝑉⊆𝑃𝑒𝑟𝑚(Id𝑉)⊆ℛ_𝑉.

Proof. First note that any identity 𝑥_𝑗≈𝑥_𝑘 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 let 𝑠≈𝑡 be an identity of 𝑉, with 𝜋 and 𝜌 permutations on the appropriate sets. We saw above that 𝑠ℛ_𝑉𝑡, and by Proposition 3.6 also 𝑠ℛ_𝑉𝜋(𝑠) and 𝑡ℛ_𝑉𝜌(𝑡). By the symmetry and transitivity of ℛ_𝑉 we get 𝜋(𝑠)ℛ_𝑉𝜌(𝑡). This shows that 𝑃𝑒𝑟𝑚(Id𝑉)⊆ℛ_𝑉.

We note that as a consequence of Proposition 3.8, the equivalence relation ℛ_𝑉 is not in general an equational theory on 𝑊_𝜏(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 consider 𝑉=𝐴𝑙𝑔(𝜏), the variety of all algebras of type 𝜏. It is well-known that for this variety 𝑉, Id𝑉=Δ_{𝑊𝜏(𝑋)}, the identity relation on 𝑊_𝜏(𝑋); that is, an identity 𝑠≈𝑡 holds in 𝑉 if and only if 𝑠=𝑡. From Proposition, we know that 𝑃𝑒𝑟𝑚(Δ_{𝑊𝜏(𝑋)}) is a subset of ℛ_𝑉, 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 𝑠ℛ_𝑉𝑡. Without loss of generality, let us assume that 𝑛≥𝑚. Then there exist terms 𝑡₁,…,𝑡_𝑚 and 𝑠₁,…,𝑠_𝑛 in 𝑊_𝜏(𝑋) such that𝑠≈𝑆^𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑚∈Id𝑉,𝑡≈𝑆^𝑛𝑚𝑠,𝑠¹,…𝑠_𝑛∈Id𝑉.(3.3) The property that Id𝑉=Δ_{𝑊𝜏(𝑋)} means that𝑠=𝑆^𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑚,𝑡=𝑆^𝑛𝑚𝑠,𝑠¹,…𝑠_𝑛.(3.4) Then we have
𝑠==𝑆𝑠_𝑚𝑛𝑥₁,…,𝑥_𝑛=𝑆^𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑚=𝑆_𝑛𝑛𝑆^𝑛𝑚(𝑠,𝑠₁,…,𝑠_𝑛,𝑡¹,…,𝑡_𝑚𝑠,𝑆^𝑚𝑛𝑠¹,𝑡₁,…,𝑡_𝑚,…,𝑆^𝑚𝑛𝑠^𝑛,𝑡₁,…,𝑡_𝑚,byC1(3.5) This equality forces a strong condition on the entries in the last line. Suppose that the variables occurring in term 𝑠 are 𝑥_𝑖1,…,𝑥_𝑖𝑘, with 𝑘≤𝑛. Then we must have 𝑆^𝑚𝑛(𝑠_𝑖𝑗,𝑡₁,…,𝑡_𝑚)=𝑥_𝑖𝑗 for each 𝑗=1,2,…,𝑘. Then for each index 𝑖_𝑗 there must exist an index 𝑙_𝑗 such that 𝑠_𝑖𝑗=𝑥_𝑙𝑗 and 𝑡_𝑖𝑗=𝑥_𝑖𝑗. Moreover the indices 𝑙_𝑗, for 1≤𝑗≤𝑘 must be distinct. This means that there is a permutation 𝜋 on the set {1,2,…,𝑛}, such that 𝜋(𝑖_𝑗)=𝑙_𝑗, for 𝑞≤𝑗≤𝑘. Then we have𝑡==𝑆𝑡_𝑛𝑚𝑥₁,…,𝑥_𝑚=𝑆^𝑛𝑚𝑠,𝑠¹,…,𝑠_𝑛=𝜋(𝑠),𝑠,𝑠¹,𝑥_𝑙1,…,𝑥_𝑙2,…,𝑥_𝑙𝑘,…,𝑠_𝑛(3.6) showing that we can obtain 𝑡 by variable permutation from 𝑠.

Example 3.10. A nontrivial variety 𝑉 of type 𝜏 is said to be normal if it does not satisfy any identity of the form 𝑥_𝑗≈𝑡, where 𝑥_𝑗 is a variable and 𝑡 is a nonvariable term. For each type 𝜏, there is a smallest normal variety 𝑁_𝜏, which is defined by the set of identities {𝑠≈𝑡∣𝑠,𝑡∈𝑊_𝜏(𝑋)⧵𝑋}. That is, any two nonvariable terms are related by Id𝑁_𝜏, while each variable is related only to itself. Using the fact that (𝑋×𝑋)∪Id𝑉 is always contained in ℛ_𝑉, we see that ℛ_𝑁𝜏=(𝑋×𝑋)∪𝑊_𝜏(𝑋)²=𝑃𝑒𝑟𝑚(Id𝑁_𝜏). This gives another example of a variety 𝑉 for which ℛ_𝑉=𝑃𝑒𝑟𝑚(Id𝑉).

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

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

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

4. The Relation ℛ_𝑉 for Varieties of Semigroups

In this section we describe the relations ℛ_𝑉 and ℒ_𝑉 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 term 𝑓(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=𝑥^𝑎 and 𝑡=𝑥^𝑏 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].

Let 𝑠=𝑠(𝑥₁,…,𝑥_𝑛) be a term of some arity 𝑛≥1, and let 𝜋 be a permutation from 𝑆𝑦𝑚_𝑚 for some 𝑚≥𝑛. In Section 3 we defined 𝜋(𝑠) to be the term 𝑠(𝑥_𝜋(𝑥1),…,𝑥_{𝜋(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. Let 𝑉 be a variety of semigroups. If 𝑉 is regular, then 𝑃𝑒𝑟𝑚(Id𝑉)is permutation-regular.

We saw in Section 3 that any two terms of the same arity 𝑛≥2 are ℒ_𝑉-related, for any variety 𝑉, and that only terms of the same arity can be ℒ_𝑉-related. Thus the only thing of interest for ℒ_𝑉 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={𝑥^𝑖|𝑖≥1}.

Proposition 4.2. For any variety 𝑉 of semigroups, ℒ_𝑉∩𝑇²¹=ℛ_𝑉∩𝑇²¹. That is, two unary terms are ℒ_𝑉-related if and only if they are ℛ_𝑉-related.

Proof. Let 𝑥^𝑖 and 𝑥^𝑗 be two unary terms, for 𝑖,𝑗≥1, with 𝑖≠𝑗. Then 𝑥^𝑖ℛ_𝑉𝑥^𝑗 if and only if 𝑥^𝑖≈𝑆¹¹(𝑥^𝑗,𝑥^𝑝) and 𝑥^𝑗≈𝑆¹¹(𝑥^𝑖,𝑥^𝑞) both hold in Id𝑉, for some unary terms 𝑥^𝑝 and 𝑥^𝑞. These identities hold if and only if 𝑥^𝑖≈𝑥^𝑗𝑝 and 𝑥^𝑗≈𝑥^𝑖𝑞 hold in 𝑉. Similarly, 𝑥^𝑖ℒ_𝑉𝑥^𝑗 if and only if 𝑥^𝑖≈𝑆¹¹(𝑥^𝑝,𝑥^𝑗) and 𝑥^𝑗≈𝑆¹¹(𝑥^𝑞,𝑥^𝑖) both hold in Id𝑉, for some unary terms 𝑥^𝑝 and 𝑥^𝑞, which is also equivalent to having both 𝑥^𝑖≈𝑥^𝑗𝑝 and 𝑥^𝑗≈𝑥^𝑖𝑞 in Id𝑉.

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

Corollary 4.3. Let 𝑉 be a variety of semigroups and let 𝑥^𝑖 and 𝑥^𝑗 be unary terms with 𝑖≠𝑗. Then 𝑥^𝑖ℛ_𝑉𝑥^𝑗 if and only if the identities 𝑥^𝑖≈𝑥^𝑝𝑗 and 𝑥^𝑗≈𝑥^𝑞𝑖 hold in 𝑉 for some natural numbers 𝑝, 𝑞≥1.

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

Proposition 4.4. Let 𝑉 be a variety of semigroups which is not uniformly periodic. Then ℒ_𝑉∩𝑇²¹=ℛ_𝑉∩𝑇²¹=Δ_𝑇1; that is, two unary terms are related by ℛ_𝑉 if and only if they are equal.

Proof. Let 𝑥^𝑖 and 𝑥^𝑗 be two unary terms which are ℒ_𝑉- or ℛ_𝑉-related. By Corollary 4.3, this forces identities of the form 𝑥^𝑖≈𝑥^𝑝𝑗 and 𝑥^𝑗≈𝑥^𝑞𝑖 to hold in 𝑉, for some natural numbers 𝑝 and 𝑞. But when 𝑉 is not uniformly periodic, an identity of the form 𝑥^𝑎≈𝑥^𝑏 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 𝑥^𝑖 and 𝑥^𝑗 are in fact equal.

What happens with unary terms for uniformly periodic varieties depends on the particular variety. We recall from Section 3 that 𝑃𝑒𝑟𝑚(Id𝑉)⊆ℛ_𝑉. We will show that if 𝑉 is both regular and uniformly periodic, then Id𝑉∩𝑇²¹=𝑃𝑒𝑟𝑚(Id𝑉)∩𝑇²¹, but ℛ_𝑉∩𝑇²¹ can be larger.

Lemma 4.5. If 𝑉 is a variety of semigroups which is both regular and uniformly periodic, then Id𝑉∩𝑇²¹=𝑃𝑒𝑟𝑚(Id𝑉)∩𝑇²¹.

Proof. Since Id𝑉⊆𝑃𝑒𝑟𝑚(Id𝑉) by definition, we know that Id𝑉∩𝑇²¹⊆𝑃𝑒𝑟𝑚(Id𝑉)∩𝑇²¹. For the opposite inclusion, suppose that 𝑥^𝑖≈𝑥^𝑗 is in 𝑃𝑒𝑟𝑚(Id𝑉) for some unary terms 𝑥^𝑖 and 𝑥^𝑗. Then there exist some identity 𝑠≈𝑡 in Id𝑉 and some permutations 𝜋 and 𝜌 such that 𝑥^𝑖=𝜋(𝑠) and 𝑥^𝑗=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 𝑥^𝑖𝑘 and 𝑥^𝑗𝑚, respectively, for some variables 𝑥_𝑘 and 𝑥_𝑚. Since 𝑉 is regular and 𝑠≈𝑡 is in Id𝑉, the variables 𝑥_𝑘 and 𝑥_𝑚 must in fact be the same. Therefore 𝑥^𝑖≈𝑥^𝑗 is actually in Id𝑉.

Any uniformly periodic variety 𝑉 must satisfy an identity of the form 𝑥^𝑎≈𝑥^𝑎+𝑏 for some natural numbers 𝑎 and 𝑏. We denote by 𝐵_𝑎,𝑏 the variety Mod{𝑥(𝑦𝑧)≈(𝑥𝑦)𝑧,𝑥^𝑎≈𝑥^𝑎+𝑏}, known as a Burnside variety. Thus any uniformly periodic variety of semigroups is a subvariety of 𝐵_𝑎,𝑏 for some 𝑎,𝑏≥1. An important fact about the identities of the variety 𝐵_𝑎,𝑏 is the following: an identity of the form 𝑥^𝑢≈𝑥^𝑣 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 ℛ_𝑉-related for the variety 𝑉=𝐵_𝑎,𝑏.

Corollary 4.6. Let 𝑉=𝐵_𝑎,𝑏, for 𝑎,𝑏≥1. Then 𝑥^𝑖ℛ_𝑉𝑥^𝑗 if and only if both 𝑖,𝑗≥𝑎 and the congruences 𝑖𝑝≡𝑗 modulo 𝑏 and 𝑗𝑞≡𝑖 modulo 𝑏 have solutions 𝑝,𝑞≥1.

Some basic number theory now provides us with some examples. Let us note that in 𝑉=𝐵_𝑎,𝑏, the unary terms are (up to equivalence modulo Id𝑉 and hence equivalence in ℛ_𝑉 as well) 𝑥,𝑥²,…,𝑥^{𝑎+𝑏−1}. In the case 𝑎=𝑏=1, we have all unary terms equivalent, and ℛ_𝑉∩𝑇²¹=𝑇²¹. For 𝑉=𝐵_𝑎,1 or 𝑉=𝐵_𝑎,2, for any 𝑎≥1, it is easy to see that ℛ_𝑉∩𝑇²¹ is just Id𝑉∩𝑇²¹. But for 𝑉=𝐵_1,𝑎 when 𝑎 is a prime number, the terms 𝑥,𝑥²,…,𝑥^𝑎−1 are all ℛ_𝑉-related to each other, but not to 𝑥^𝑎; in this case more terms are related by ℛ_𝑉 than those related by Id𝑉. For 𝑉=𝐵_2.5, we can show that there are 3 distinct classes of terms under ℛ_𝑉: {𝑥}, {𝑥²,𝑥³,𝑥⁴,𝑥⁶} and {𝑥⁵}. This shows that for this choice of 𝑉, we have 𝑃𝑒𝑟𝑚(Id𝑉)⊂ℛ_𝑉⊂∇_{𝑊𝜏(𝑋)}.

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

Proposition 4.7. If 𝑉 is a variety of semigroups which is not uniformly periodic, then ℛ_𝑉=𝑃𝑒𝑟𝑚(Id𝑉).

Proof. This proof is a modification of the argument from Example 3.9. First, by Proposition 3.8 we have 𝑃𝑒𝑟𝑚(Id𝑉)⊆ℛ_𝑉, 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 𝑠ℛ_𝑉𝑡. Then there exist terms 𝑡₁,…,𝑡_𝑚 and 𝑠₁,…,𝑠_𝑛 in 𝑊_𝜏(𝑋) such that𝑠≈𝑆^𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑚∈Id𝑉,𝑡≈𝑆^𝑛𝑚𝑠,𝑠¹,…𝑠_𝑛∈Id𝑉.(4.1) Then we have𝑠≈≈𝑆𝑆^𝑚𝑛_𝑚𝑛𝑡,𝑡¹,…,𝑡_𝑚≈𝑆_𝑛𝑛𝑆^𝑛𝑚(𝑠,𝑠₁,…,𝑠_𝑛,𝑡¹,…,𝑡_𝑚𝑠,𝑆^𝑚𝑛𝑠¹,𝑡₁,…,𝑡_𝑚,…,𝑆^𝑚𝑛𝑠^𝑛,𝑡₁,…,𝑡_𝑚,byC1(4.2) 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 𝑆^𝑛𝑛(𝑠,𝑆^𝑚𝑛(𝑠₁,𝑡₁,…,𝑡_𝑚),…,𝑆^𝑚𝑛(𝑠_𝑛,𝑡₁,…,𝑡_𝑚)) 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 𝑥_𝑖1,…,𝑥_𝑖𝑘, with 𝑘≤𝑛. Then we must have 𝑆^𝑚𝑛(𝑠_𝑖𝑗,𝑡₁,…,𝑡_𝑚)=𝑥_𝑖𝑗 for each 𝑗=1,2,…,𝑘. Then for each index 𝑖_𝑗 there must exist an index 𝑙_𝑗 such that 𝑠_𝑖𝑗=𝑥_𝑙𝑗 and 𝑡_𝑖𝑗=𝑥_𝑖𝑗. Moreover the indices 𝑙_𝑗, for 1≤𝑗≤𝑘 must be distinct. This means that there is a permutation 𝜋 on the set {1,2,…,𝑛}, such that 𝜋(𝑖_𝑗)=𝑙_𝑗, for 𝑞≤𝑗≤𝑘. Then we have𝑡==𝑆𝑡_𝑛𝑚𝑥₁,…,𝑥_𝑚=𝑆^𝑛𝑚𝑠,𝑠¹,…,𝑠_𝑛=𝜋(𝑠).𝑠,𝑠¹,𝑥_𝑙1,…,𝑥_𝑙2,…,𝑥_𝑙𝑘,…,𝑠_𝑛(4.3)This shows that 𝑡=𝜋(𝑠) for some permutation 𝜋, and hence that ℛ_𝑉⊆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 ℛ_𝑉 for this variety 𝑉 is equal to 𝑃𝑒𝑟𝑚(Id𝑉), from Example 3.10.

At the other extreme is the variety 𝐵_1,1 of idempotent semigroups or bands. The lattice ℒ(𝐵_1,1) 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 ℛ_𝑉 is the total relation ∇_{𝑊𝜏(𝑋)} on 𝑊_𝜏(𝑋).

Theorem 4.8. Let 𝑉 be a variety of semigroups. Then ℛ_𝑉=∇_{𝑊𝜏(𝑋)} if and only if 𝑉 is a subvariety of the variety 𝐵_1,1 of bands.

Proof. First let 𝑉 be a variety of bands, so 𝑉⊆𝐵_1,1. 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≈𝑆^𝑛𝑚(𝑠,𝑡,…,𝑡)∈Id𝑉 and 𝑠≈𝑆^𝑚𝑛(𝑡,𝑠,…,𝑠)∈Id𝑉, making 𝑠ℛ_𝑉𝑡.
Conversely, suppose that 𝑉 has the property that any two terms (of any arities) are related by ℛ_𝑉. 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=𝑥^𝑐, for some 𝑐≥1. In particular, our variety 𝑉 must satisfy an identity of the form 𝑥≈𝑥^𝑎 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≈𝑥^𝑎 :𝑥𝑥≈𝑥^𝑎≈𝑥^2𝑎−1≈𝑥^3𝑎−2≈⋯∈Id𝑉₂𝑥≈𝑥^𝑎+1≈𝑥^2𝑎≈𝑥^3𝑎−1≈⋯∈Id𝑉₃𝑥⋮⋮_𝑎−1≈𝑥^𝑎+2≈𝑥^2𝑎+1≈𝑥^3𝑎≈⋯∈Id𝑉≈𝑥^2(𝑎−1)≈𝑥^3(𝑎−1)≈𝑥^4(𝑎−1)≈⋯∈Id𝑉.(4.4) Now we also know that 𝑥 is ℛ_𝑉-related to 𝑥^𝑎−1, which means that we can write 𝑥≈𝑆¹¹(𝑥^𝑎−1,𝑞)∈Id𝑉 for some unary term 𝑞=𝑥^𝑘, for some 𝑘. Therefore, we get 𝑥≈𝑥^{𝑘(𝑎−1)}∈Id𝑉. A similar argument applied to 𝑥²ℛ_𝑉𝑥^𝑎−1 then gives 𝑥²≈𝑥^{𝑚(𝑎−1)}∈Id𝑉 for some 𝑚. Since 𝑥^{𝑘(𝑎−1)}≈𝑥^{𝑚(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. Let 𝑉=𝐵_𝑎,𝑏 for some 𝑎,𝑏≥1. Let 𝑡 be any term of arity 𝑛≥2 which has at least one variable 𝑥_𝑘 occurring in it a number of times which is congruent to 1 modulo 𝑏. Then 𝑥^𝑎ℛ_𝑉𝑡^𝑎.

Proof. We can always write 𝑡^𝑎=𝑆^1𝑛(𝑎^𝑎,𝑝) for some 𝑛-ary term 𝑝, by taking 𝑝=𝑡. But we also need to be able to write 𝑥^𝑎≈𝑆^𝑛1(𝑡^𝑎,𝑞₁,…,𝑞_𝑛)∈Id𝑉 for some unary terms 𝑞₁,…,𝑞_𝑛. Let 𝑥_𝑘 be a variable which occurs in 𝑡 exactly 𝑣 times, where 𝑣 is congruent to 1 modulo 𝑏. For the term 𝑞_𝑘, we use 𝑥, and for all the other terms 𝑞₁,…,𝑞_𝑛, we use 𝑥^𝑏. Then 𝑆^𝑛1(𝑡^𝑎,𝑞₁,…,𝑞_𝑛)=𝑡^𝑎(𝑥^𝑏,…,𝑥^𝑏,𝑥,𝑥^𝑏,…,𝑥^𝑏)=(𝑥^𝑞𝑏+1)^𝑎 for some natural number 𝑞. Then in 𝐵_𝑎,𝑏 we have (𝑥^𝑞𝑏+1)^𝑎≈𝑥^{𝑎𝑞𝑏+𝑎}≈𝑥^𝑎, as required.

Corollary 4.10. Let 𝑉=𝐵_𝑎,𝑏 for some 𝑎,𝑏≥1 with 𝑎+𝑏≥3. Then 𝑃𝑒𝑟𝑚(Id𝑉) is a proper subset of ℛ_𝑉, which is a proper subset of ∇_{𝑊𝜏(𝑋)}on𝑊_𝜏(𝑋).

Proof. By the previous theorem, we have 𝑥^𝑎ℛ_𝑉(𝑥𝑦)^𝑎. Since the terms 𝑥^𝑎 and (𝑥𝑦)^𝑎 contain different numbers of variables, and 𝑉 is regular, the identity 𝑥^𝑎≈(𝑥𝑦)^𝑎 cannot be in 𝑃𝑒𝑟𝑚(Id𝑉). Thus 𝑃𝑒𝑟𝑚(Id𝑉) is a proper subset of ℛ_𝑉. The remaining claim follows from Theorem 4.8.

Acknowledgment

This research is supported by the NSERC of Canada.