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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

A semigroup is an algebra of type

To define Green's relations on a semigroup

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

We begin with some notation. Throughout this paper, we
will assume a type

As a preliminary step in defining Green's relations on
any algebra of arbitrary type, let us consider first the case of type

Let

If the type

Now consider an algebra

Let

Again these two relations are clearly seen to be
reflexive and symmetric on the base set

A

This gives us the algebra

Let

For any type

As noted above, both relations

Transitivity of

A similar definition of a Green's-like relation

Now, we extend our definition of Green's relations

Let

This definition actually includes Definition

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

In this section, we describe some properties of the
relations

two terms of type

the relation

(i)
It follows
from the definition of superposition of terms that the term

(ii) For any variety

We have shown that any two terms of the same arity

This was
proved in [

Let

Next we consider the right relation

(i)

(ii) If

(i) Follows immediately from the fact that

Now we want to prove some facts about which pairs of
terms can be

Let

To further describe

By definition

Let

First note that any identity

Now let

We note that as a consequence of Proposition

In this
example we consider

A nontrivial variety

We can use the relation

When

In this section we describe the relations

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

Let

We saw in Section

Let

This result allows us to completely characterize the
relation

Now we describe how the relations

Let

What happens with unary terms for uniformly periodic
varieties depends on the particular variety. We recall from Section

If

Since

Any uniformly periodic variety

Some basic number theory now provides us with some
examples. Let us note that in

Finally, we consider the relation

This proof is a modification of the argument from
Example

The converse of this proposition is not however true.
As an example we consider the smallest normal variety of type

At the other extreme is the variety

First let

Conversely, suppose that

We can always write

By the previous theorem, we have

This research is supported by the NSERC of Canada.