Commutative orders

David Easdown and Victoria Gould

December 1995

School of Mathematics and Statistics Department of Mathematics University of Sydney University of York Sydney NSW 2006 Heslington, York YO1 5DD Australia England

Abstract

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a^{∗}b(q = ba^{∗}) for some a, b ∈ S, where a^{∗} denotes the inverse of a in a subgroup of Q and if, in addition, every squarecancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity x_{1}...x_{n} = x_{1π}...x_{nπ} where 1 < 1π and nπ < n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.

1 Introduction

It is well known that a semigroup S has a group of left quotients if and only if S is cancellative and right reversible, that is, Sa ∩ Sb ≠ ∅ for all a, b ∈ S [CP, Theorem 1.24]. Thus a commutative semigroup S has a group G of left quotients if and only if S is cancellative; in this case it is easy to see that G is also commutative. We are concerned in this paper with the more general notion of a semigroup of left quotients. The concept we use is that introduced by Fountain and Petrich in [3]; the main idea is that we consider inverses of elements in any subgroup of a semigroup, and not just the group of units.

Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every q ∈ Q can be written as q = a^{∗}b for some a, b ∈ S, where a^{∗} denotes the inverse of a in a subgroup of Q and if, in addition, every element of S satisfying a weak cancellation condition known as squarecancellability lies in a subgroup of Q. Clearly if S has a group of left quotients G then G is also a semigroup of left quotients of S. If Q is a semigroup of left quotients of S we also say that S is a left order in Q. Semigroups of right quotients and right orders are defined dually. If S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S.

It is natural to hope that if a commutative semigroup S is a left order in Q, then Q is also commutative so that S is an order in Q. This is true; indeed if S satisfies any permutation identity x_{1}...x_{n} = x_{1π}...x_{nπ} where 1 < 1π and nπ < n, then Q is commutative, as we show in Theorem 3.1. Semigroups satisfying a permutation identity have attracted interest from a number of authors; see, for example, [10] and [7]. We remark that it is shown in [7] that if a semigroup S satisfies a permutation identity of the aforementioned kind and S^{2} = S, that is, S is globally idempotent, then S is commutative. Our left orders, however, are not necessarily globally idempotent.

A general description of semigroups that are left orders is not known and would undoubtedly be unwieldy. Authors have therefore concentrated on studying semigroups that are (left) orders in semigroups in a particular class, for example, orders in completely 0-simple semigroups are characterised in [3]. Surprisingly, orders in commutative semigroups have not been studied, a situation we hope to amend in this paper. In particular we give a description of orders in commutative semigroups. In view of Theorem 3.1, this is the class of commutative orders.

Section 2 contains definitions and preliminary remarks on orders. Section 3 concentrates on proving the above mentioned result that left orders satisfying certain permutation identities, and their semigroups of (left) quotients, are commutative. Then in Section 4 we give the promised description of commutative orders. Theorems 4.2 and 4.3 show that if S is a commutative semigroup then the existence of a semigroup of quotients of S is dependent upon the existence of a preorder on S satisfying certain conditions. The versatility of this result is illustrated in Examples 7.4 and 7.5, where it is used to construct semigroups of quotients having various prescribed properties.

An example is given in [4] of a commutative semigroup having nonisomorphic semigroups of quotients. With this in mind we determine in Section 5 when one semigroup of quotients of a commutative semigroup S is a homomorphic image of another. As a corollary, we can decide when two semigroups of quotients of S are isomorphic. We also show that for certain commutative orders S, namely those in which all elements are squarecancellative, the set of semigroups of quotients of S forms a complete lattice under a natural partial order. Such orders S have a maximum and a minimum semigroup of quotients.

In Section 6 we study the situation where a commutative semigroup S is a semilattice Y of semigroups S_{α}, α ∈ Y, where S_{α} is an order in Q_{α} for each α ∈ Y. We give necessary and sufficient conditions for S to be an order in Q, where Q is a semilattice Y of semigroups Q_{α}, α ∈ Y. As a consequence of this we can show that if S is a semilattice Y of commutative cancellative semigroup S_{α}, α ∈ Y, then S is an order in a commutative regular semigroup. This corollary is also known from [4], which studies left orders in regular semigroups with central idempotents.

In our final section we give a number of examples to illustrate our results. Examples 7.1 and 7.2 are examples of commutative orders where not all elements are square-cancellable, but such that the set of semigroups of quotients of each forms a complete lattice. They also show that the preorder in Theorem 4.2 cannot always be replaced by the preorder ≤_{H^{∗}} (unlike the case where all elements of an order are square-cancellable). Examples 7.4 (7.5) are, respectively, commutative orders having a maximum but no minimum (a minimum but no maximum) semigroup of quotients.

2 Preliminaries

We assume the reader has some knowledge of algebraic semigroup theory, in particular the definitions and elementary facts concerning Green’s relations. Any undefined notation or concepts may be found in the standard references [1] or [6]. We deviate from standard notation in denoting by a^{∗} the group inverse, where it exists, of an element a of a semigroup Q. That is, a^{∗} exists if and only if a lies in a subgroup of Q, and the inverse of a in this subgroup is a^{∗}. By a famous result of Green [H, Theorem II 2.5], a^{∗} exists if and only if a is related to its square by the relation H. Moreover, where a^{∗} exists it is unique. We write H(Q) for the union of the subgroups of Q. That is, H(Q) = {a : aHa^{2}}. In general, H(Q) will not be a subsemigroup.

To define square-cancellability we make use of a generalisation of Green’s relations. Let a, b be elements of a semigroup S. The relation L^{∗} is defined by the rule that aL^{∗}b if and only if aLb in some oversemigroup of S. This is equivalent to the cancellation condition that

ax = ay if and only if bx = by

for all x, y ∈ S^{1} [2]. The relation R^{∗} is defined dually and we put H^{∗} = L^{∗} ∩ R^{∗}. It is easy to see that L^{∗} is a right congruence and R^{∗} is a left congruence. Thus if S is commutative, L^{∗} = R^{∗} = H^{∗} is a congruence on S.

An element a of a semigroup S is square-cancellable if aH^{∗}a^{2}. Squarecancellability is thus a necessary condition for an element to lie in a subgroup of an oversemigroup. The definition of a semigroup of (left) quotients insists that all such elements must lie in subgroups of any semigroup of (left) quotients. We denote by S(S) the set of square-cancellable elements of a semigroup S.

Let S be a subsemigroup of Q. Then S is a weak left order in Q if any q ∈ Q can be written as q = a^{∗}b where a, b ∈ S. If in addition H(Q) ∩ S = S(S), then S is a left order in Q and Q is a semigroup of left quotients of S. The left-right dual and the two-sided notions are defined in the obvious way.

If S is a weak left order in Q, then by definition any q ∈ Q may be written as q = a^{∗}b where a, b ∈ S. Hence q = (a^{2})^{∗}ab and abQ^{1} ⊆ aQ^{1} = a^{2}Q^{1}. Thus any q ∈ Q can be written as q = c^{∗}d where c, d ∈ S and d≤_{R}c in Q. Here ≤_{R} is the preorder given by

u≤_{R}v if and only if uQ^{1} ⊆ vQ^{1}

where u, v ∈ Q. The preorder ≤_{L} is defined dually; ≤_{R} (≤_{L}) is left (right) compatible with multiplication and has associated equivalence relation R (L). Where Q is commutative ≤_{R} = ≤_{L} and we denote this relation by ≤_{H}. Also from the definition, we have that if S is a weak left order in Q then given any q ∈ Q, there is an a ∈ S(S) with q≤_{R}a in Q; from this, q = aa^{∗}q and Qq = Q^{1}q. Further, if q = a^{∗}b where a, b ∈ S and b≤_{R}a, then qLb in Q. Thus S has non-empty intersection with every L-class of Q; if Q is commutative, qHb so that S has non-empty intersection with every H-class of Q. We state these facts as a lemma, which we use repeatedly.

Lemma 2.1 Let S be an order in a commutative semigroup Q. Then any q ∈ Q can be written as q = a^{∗}b where a, b ∈ S, b≤_{H}a and qHb in Q. In particular, S has non-empty intersection with every H-class of Q.

In Section 3 we show that if S is commutative and a weak left order in Q, then Q is commutative. The results of Sections 4 and 5 show that if S is a commutative order, then a semigroup of quotients Q is determined by the preorder ≤_{H}∩(S ×S) (where ≤_{H} is the relation on Q). An obvious candidate for this preorder is ≤_{H^{∗}}, which is defined on a commutative semigroup S by the rule that a≤_{H^{∗}}b if

xb = yb implies that xa = ya

for all x, y ∈ S^{1}. As in [8], if a, b ∈ S then a≤_{H^{∗}}b if and only if a≤_{R}b in T and a≤_{L}b in U, for some oversemigroups T and U of S. Clearly ≤_{H^{∗}} is a preorder compatible with multiplication, with associated equivalence relation H^{∗}. If S is a (weak) left order in Q then Green’s relations and their preorders will always refer to relations on Q; the starred versions will refer to relations on S.

We end this section by gathering together some elementary remarks concerning the relations H^{∗} and H on a commutative semigroup.

Lemma 2.2 Let T be a commutative semigroup. Then

(i) H^{∗} is a congruence on T and S(T) is a subsemigroup of T; (ii) H is a congruence on T and H(T) is a subsemigroup, in fact H(T) is a semilattice of the group H-classes of T; (iii) for all a, b ∈ H(T), (ab)^{∗} = b^{∗}a^{∗} = a^{∗}b^{∗}. Further, if S is an order in a commutative semigroup Q, then S(S) is an order in H(Q).

3 Left orders satisfying a permutation identity

In proving our first result it is convenient to make a slight adjustment in notation. If a is an element of a semigroup Q and a lies in a subgroup of Q then for any positive integer n we write a^{−n} for (a^{∗})^{n}.

Theorem 3.1 Let S be a weak left order in Q and suppose that S satisfies a permutation identity

x_{1}...x_{n} = x_{1π}...x_{nπ} (†)

for some permutation π for which 1 ≠ 1π and n ≠ nπ. Then Q is commutative.

Proof Put k = 1π^{−1} > 1 and ℓ = nπ^{−1} < n. Observe that, for all a, b ∈ S two applications of (†) give

a^{2n−2}b = (a^{2})^{n−1}b = (a^{2})^{ℓ−1}b(a^{2})^{n−ℓ}
= (a^{2})^{ℓ−1}(ba)a(a^{2})^{n−ℓ−1} = a^{2n−3}ba.

Thus for any positive integer K and any integer M ≥ 2n − 3,

a^{M+K}b = a^{M}ba^{K}.

Suppose now that a, b ∈ S, that a lies in a subgroup of Q and b≤_{R}a. Then, since a^{−1}ab = b, the above yields

ab = a^{−2n+3}a^{2n−2}b = a^{−2n+3}a^{2n−3}ba = ba,

and so also

a^{−1}b = a^{−2}ab = a^{−2}ba = a^{−2}ba^{2}a^{−1} = a^{−2}a^{2}ba^{−1} = ba^{−1}.

Now suppose that s, t ∈ S and s lies in a subgroup of Q. Then, since s^{k}t≤_{R}s^{k−1}, from another application of (†),

st = s^{−k+1}(s^{k}t) = (s^{k}t)s^{−k+1} = s^{k−2}s^{2}ts^{n−k}s^{−n+1} = ts^{n}s^{−n+1} = ts,

and so also, as st≤_{R}s^{2},

s^{−1}t = s^{−2}(st) = (st)s^{−2} = (ts)s^{−2} = ts^{−1}.

If further t lies in a subgroup of Q then

s^{−1}t^{−1} = s^{−1}tt^{−2} = ts^{−1}t^{−2} = t^{−2}t^{3}s^{−1}t^{−2}
= t^{−2}s^{−1}t^{3}t^{−2} = t^{−2}s^{−1}t = t^{−2}ts^{−1} = t^{−1}s^{−1}.

Finally, consider h, k ∈ S. As commented in Section 2, h≤_{R}a for some element a of S lying in a subgroup of Q. Then hk = a^{−(n−2)}a^{ℓ−1}ha^{n−ℓ−1}k, since a^{−1}ah = aa^{−1}h = h, and since all powers of a commute with all elements of S. (If ℓ − 1 = 0 or n − ℓ − 1 = 0 we consider the corresponding power of a to be the empty word.) Using (†) we have that, for some integer p

hk = a^{p}ka^{−p}h = ka^{p}a^{−p}h = kh.

It follows that S and Q are commutative.

Corollary 3.2 is now immediate.

Corollary 3.2 Let S be a commutative weak left order in Q. Then Q is commutative.

If Q is a regular semigroup then clearly Q is a weak left order in itself. The following corollary appears as Theorem 6 in [10].

Corollary 3.3 [10] Let Q be a regular semigroup satisfying a permutation identity

x_{1}...x_{n} = x_{1π}...x_{nπ}

for some permutation π for which 1 ≠ 1π and n ≠ nπ. Then Q is commutative.

In fact, if Q is a regular semigroup then it is easy to see that R^{∗} = R and L^{∗} = L. Hence H^{∗} = H and H(Q) = S(Q). Thus Q is an order in itself.

4 Characterisation of commutative orders

In this section we give necessary and sufficient conditions for a commutative semigroup S to be an order. Necessary conditions are easy to obtain; as is usual with these problems, proof of sufficiency is more involved. The most general description of left orders to date is given in [5], where straight left orders are described. Commutative orders, although in many ways less complex than arbitrary orders, need not be straight, so we cannot here make use of the results of [5] or previous work on orders. Examples of non-straight commutative orders are given in Section 7.

The proof of our first lemma is straightforward.

Lemma 4.1 Let S be a commutative order in a semigroup Q and denote ≤_{H} ∩ (S × S) by ≤, where H is Green’s relation on Q. Then ≤ is a preorder compatible with multiplication such that: (A) for all b, c ∈ S, bc ≤ b; (B) for all b, c ∈ S and a ∈ S(S) ,

b ≤ a, c ≤ a, ab = ac

implies that b = c ≤ ba; (C ′) for all b, c ∈ S, b ≤ c implies that bx = cy for some x ∈ S(S), y ∈ S with b ≤ x.

For our construction proof we make use of a weaker version of (C′), namely: (C) for all b ∈ S there exists a ∈ S(S) with b ≤ a.

Theorem 4.2 Let S be a commutative semigroup. Then S is an order in some semigroup Q if and only if there exists a preorder ≤ on S which is compatible with multiplication such that conditions (A), (B) and (C) hold.

Further, Q may be chosen such that

≤_{H} ∩ (S × S) ⊆≤

and for all b ∈ S and a ∈ S(S)

b ≤ a if and only if b≤_{H}a.

Proof The necessity of the conditions is immediate from Lemma 4.1 and the comment which follows it.

Suppose conversely that S satisfies conditions (A), (B) and (C). We aim to construct a semigroup Q in which S is an order.

Let ≡ be the equivalence relation on S associated with the preorder ≤; since ≤ is compatible with multiplication, ≡ is a congruence on S. By conditions (A) and (B) (with b = c = a) we have that if a ∈ S(S) then a ≡ a^{2}. Put

∑ = {(a, b) ∈ S(S) × S : b ≤ a}

so that ∑ ≠ ∅ by (C). Define a relation ∼ on ∑ by the rule that for (a, b), (c, d) ∈ ∑,

(a, b) ∼ (c, d) if and only if ad = bc and b ≡ d.

Clearly ∼ is reflexive and symmetric. Suppose that (a, b), (c, d), (e, f) ∈ ∑ and

(a, b) ∼ (c, d) ∼ (e, f).

Then ad = bc, cf = de and b ≡ d ≡ f; since ≡ is a congruence, b ≡ f. Then

(af)c = (cf)a = (de)a = (ad)e = (bc)e = (be)c.

By (A), af ≤ f ≡ d ≤ c and be ≤ b ≡ d ≤ c so that (B) gives af = be. Thus ∼ is an equivalence relation on ∑.

Put Q = ∑ / ∼ and denote the ∼-equivalence class of (a, b) by [a, b]. Define a multiplication on Q by

[a, b][c, d] = [ac, bd],

which makes sense since by Lemma 2.2 S(S) is a subsemigroup of S. This multiplication on Q is well defined, for if (a, b) ∼ (a′, b′) and (c, d) ∼ (c′, d′), then b ≡ b′, d ≡ d′, ab′ = ba′ and cd′ = dc′. Now

(ac)(b′d′) = (ab′)(cd′) = (ba′)(dc′) = (a′c′)(bd)

and further, bd ≡ b′d′ so that (ac, bd) ∼ (a′c′, b′d′). Clearly the multiplication in Q is associative and commutative.

We now show that S is embedded in Q. For b ∈ S there exists a ∈ S(S) with b ≤ a, by (C); (A) and (B) together give that

ab ≡ b (‡),

so that ab ≤ a and (a, ab) ∈ ∑. Further, if a′ ∈ S(S) and b ≤ a′, then (a, ab) ∼ (a′, a′b). Thus θ : S → Q is well defined where bθ = [a, ab] and a ∈ S(S) is chosen with b ≤ a. It is easy to see that θ is a homomorphism. Suppose now that b, d ∈ S and bθ = dθ. Thus (a, ab) ∼ (c, cd) for some a, c ∈ S(S) with b ≤ a and d ≤ c. From the definition of ∼ and from (‡) it follows that b ≡ ab ≡ cd ≡ d. Also acd = acb and cb ≤ b ≤ a so that cd ≤ a and (B) gives that cd = cb. Another application of (B) gives that d = b, hence θ : S → Q is an embedding.

We now verify that Sθ is an order in Q. If a ∈ S(S) then aθ = [a, a^{2}]. Note also that (a^{2}, a) ∈ ∑ and (a, a) ∈ ∑. It is easy to check that aθ lies in a subgroup of Q with identity [a, a] and (aθ)^{∗} = [a^{2}, a]. Finally, suppose that [a, b] ∈ Q. Then

(aθ)^{∗}bθ = [a^{2}, a][a, ab] = [a^{3}, a^{2}b] = [a, b],

which completes the proof that Sθ is an order in Q.

To prove the last assertions of the theorem, let b, d ∈ S and suppose that bθ≤_{H}dθ in Q. Then there exists [x, y] ∈ Q with

[a, ab] = [c, cd][x, y] = [cx, cdy]

where a, c ∈ S(S), b ≤ a and d ≤ c. This gives that b ≡ ab ≡ cdy ≤ d so that b ≤ d. Further, given m ∈ S and n ∈ S(S) with m ≤ n, it is easy to check that

[n, n^{2}][n, m] = [n, nm]

giving mθ≤_{H}nθ.

We now show that by strengthening condition (C) in Theorem 4.2 Q may be chosen so that ≤_{H} ∩(S ×S) =≤, where ≤ is the given preorder on S. The significance of this becomes apparent in the next section where we consider conditions under which one semigroup of quotients of S is a homomorphic image of another.

Theorem 4.3 Let S be a commutative semigroup. Then S is an order in a semigroup Q such that ≤_{H} ∩ (S × S) =≤ if and only if ≤ is a preorder on S compatible with multiplication satisfying conditions (A),(B) and (C ′).

Proof The necessity of the conditions is immediate from Lemma 4.1.

Conversely, suppose that ≤ is a preorder on S compatible with multiplication such that conditions (A), (B) and (C′) hold. By Theorem 4.1, S is an order in a semigroup Q such that

≤_{H} ∩ (S × S) ⊆≤

and for all b ∈ S and a ∈ S(S)

b ≤ a if and only if b≤_{H}a.

Suppose that b, c ∈ S and b ≤ c. By condition (C′) there exists x ∈ S(S), y ∈ S with b ≤ x and bx = cy. Since x ∈ S(S) we have b≤_{H}x and b = bxx^{∗} = cyx^{∗} so that b≤_{H}c. Thus ≤_{H} ∩ (S × S) =≤.

If S is a commutative order in Q, then ≤_{H} ∩ (S × S) ⊆ ≤_{H^{∗}}; in the case where S = S(S) we show that S has some semigroup of quotients in which ≤_{H} ∩ (S × S) = ≤_{H^{∗}}.

Corollary 4.4 Let S be a commutative semigroup with S = S(S). Then S is an order if and only if the H^{∗}-classes of S are cancellative. In this case, S has a semigroup of quotients in which ≤_{H} ∩ (S × S) = ≤_{H^{∗}}.

Proof Note that for a commutative semigroup S with S = S(S), conditions (A) and (C′) always hold for ≤_{H^{∗}}. If S is an order in Q and a, b, c are H^{∗}- related elements of S with ab = ac, then b^{2} = bc = c^{2} so that bHc in Q and b = c since H_{b} is a group. Thus the H^{∗}-classes of S are cancellative.

Conversely, suppose that the H^{∗}-classes of S are cancellative and a, b, c ∈ S with

b≤_{H^{∗}}a, c≤_{H^{∗}}a and ab = ac.

Since S = S(S) we have bH^{∗}b^{2}≤_{H^{∗}}ba≤_{H^{∗}}b so that bH^{∗}ba. Similarly, cH^{∗}ca so that bH^{∗}c and from the fact that the H^{∗}-class of b is cancellative, (ba)b = (ba)c yields b = c. Thus condition (B) holds. Theorem 4.3 says that S is an order in some semigroup where ≤_{H} ∩ (S × S) = ≤_{H^{∗}}.

If a commutative semigroup S with S(S) = S is an order in Q then it is not necessarily the case that ≤_{H} ∩ (S × S) = ≤_{H^{∗}}. For example, a commutative cancellative semigroup (here H^{∗} is universal) can be an order in a non-trivial semilattice of groups (see [4]). In the final section we give examples of commutative orders for which the relation H^{∗} does not satisfy conditions (A),(B) and (C′). By Theorem 4.3, such an order S cannot have a semigroup of quotients in which ≤_{H} ∩ (S × S) = ≤_{H^{∗}}.

5 Maximum and minimum semigroups of quotients

As remarked in the introduction, a commutative semigroup S may have nonisomorphic semigroups of quotients. In this section we determine when two semigroups of quotients of S are isomorphic; this is a corollary of our first result which gives necessary and sufficient conditions for one semigroup of quotients of S to be a homomorphic image of another, under a homomorphism which restricts to the identity on S. Such a homomorphism is called an S-homomorphism. More precisely, if S is an order in Q_{1} and Q_{2} where ι : S → Q_{1} and τ : S → Q_{2} are the embeddings of S in Q_{1} and Q_{2} respectively, then a homomorphism θ : Q_{1} → Q_{2} is an S-homomorphism if ιθ = τ. If θ is a bijective S-homomorphism then we say that Q_{1} and Q_{2} are isomorphic over S.

Theorem 5.1 Let S be a commutative semigroup and an order in semigroups Q_{1} and Q_{2}. The following are equivalent:

(i) there is an onto S-homomorphism θ : Q_{1} → Q_{2}; (ii) for all a, b ∈ S

a≤_{H}b in Q_{1} implies a≤_{H}b in Q_{2};

(iii) for all a, b ∈ S

aHb in Q_{1} implies aHb in Q_{2}.

Moreover, if the above conditions hold then θ is uniquely defined.

Proof The first two implications are immediate; we prove (iii) implies (i). To clarify the notation we denote the inverse of a ∈ S in a subgroup of Q_{2} by a^{♯}, using the usual notation a^{∗} for the inverse of a in a subgroup of Q_{1}. Since S is an order in both Q_{1} and Q_{2}, a^{∗} exists if and only if a ∈ S(S) if and only if a^{♯} exists.

Suppose that (iii) holds. If b ∈ S and a ∈ S(S), then b≤_{H}a in Q_{1} implies bHba in Q_{1} so that by hypothesis bHba and b≤_{H}a in Q_{2}. Define θ : Q_{1} → Q_{2} by (a^{∗}b)θ = a^{♯}b where b≤_{H}a in Q_{1}. It is straightforward to show that θ is an S-homomorphism from Q_{1} onto Q_{2} so that (i) holds.

Now if ϕ is any S-homomorphism from Q_{1} to Q_{2} then for all a^{∗}b ∈ Q_{1}, (a^{∗}b)ϕ = (aϕ)^{♯}bϕ = a^{♯}b = (a^{∗}b)θ.

We can now determine immediately when two semigroups of quotients of S are isomorphic over S.

Corollary 5.2 Let S be a commutative semigroup and an order in Q_{1} and Q_{2}. Then Q_{1} is isomorphic to Q_{2} over S if and only if H^{Q_{1}} ∩ (S × S) = H^{Q_{2}} ∩ (S × S).

Let S be a commutative order. A semigroup of quotients Q of S is maximum if, given any semigroup of quotients P of S, there is an S-homomorphism from Q onto P. Dually, Q is minimum if, given any semigroup of quotients P of S, there is an S-homomorphism from P onto Q. Clearly, a maximum (minimum) semigroup of quotients of S is unique up to isomorphism over S. In view of Corollary 4.4 and the definition of H^{∗}, if S = S(S) then a minimum semigroup of quotients of S exists. Using Theorem 3.1 of [4] we can improve this result considerably.

At this point we note that if S is a commutative order, then since each semigroup of quotients of S is a homomorphic image of the free semigroup on T ∪ S where T ∩ S = ∅ and T is in one-one correspondence with S(S), the isomorphism classes (over S) of the semigroups of quotients of S form a set.

Proposition 5.3 Let S be a commutative order for which S(S) = S and let Q be the set of semigroups of quotients of S. Define a relation ⪯ on Q by the rule that Q ⪯ P if there is an S-homomorphism from P onto Q. Then Q is a complete lattice under ⪯. In particular, S has maximum and minumum semigroups of quotients.

Proof It is implicit in the proof of Theorem 3.1 of [4] that a commutative semigroup T is an order in a semilattice Y of (commutative) groups G_{α}, α ∈ Y, if and only if T is a semilattice Y of (commutative) cancellative semigroups T_{α} where T_{α} is an order in G_{α}, α ∈ Y. This result is also obtained as a corollary of Theorem 4.3 in the next section of this paper.

If S is an order in Q then, using the facts that S = S(S) and S has nonempty intersection with every H-class of Q, it follows from Theorem 3.1 that Q is a commutative regular semigroup, hence a semilattice of commutative groups.

By Corollary 4.4, H^{∗} is a semilattice congruence on S, all of whose classes are cancellative. Now if S is an order in Q then as Q is a semilattice of groups, H^{Q}∩(S×S) is a semilattice congruence contained in H^{∗}. Moreover, if ρ is any semilattice congruence contained in H^{∗}, then each ρ-class is a cancellative semigroup and S is a semilattice of its ρ-classes. Thus S is an order in Q_{ρ}, where H^{Q_{ρ}} ∩ (S × S) = ρ. It follows from Corollary 5.2 that there is a bijection between C and Q, where C is the set of semilattice congruences on S contained in H^{∗}. Further, if ρ, μ ∈ C then by Theorem 5.1, ρ ⊆ μ if and only if there is an S-homomorphism from Q_{ρ} onto Q_{μ}, that is, Q_{μ} ⪯ Q_{ρ}. Since C is a complete lattice, so also is Q.

If we relax the condition on a commutative order S that S = S(S) then it is not always the case that S has a maximum or minimum semigroup of quotients as we show in the final section. We also give examples of commutative orders S where S ≠ S(S) such that the semigroups of quotients of S form a complete lattice.

We finish this section with a necessary and sufficient condition for a commutative order to have a maximum semigroup of quotients. The idea for the construction was suggested to us by P.N. Anh.

Proposition 5.4 Let S be a commutative order. Then S has a maximum semigroup of quotients if and only if for every s ∈ S there exists a ∈ S(S) with s≤_{H}a in every semigroup of quotients of S.

Proof Suppose that S has a maximum semigroup of quotients P. If s ∈ S then s≤_{H}a in P for some a ∈ S(S); now if S is an order in Q then there is an S-homomorphism θ : P → Q so that sθ≤_{H}aθ in Q, that is, s≤_{H}a in Q.

Conversely, suppose that for each s ∈ S there exists a ∈ S(S) such that s≤_{H}a in every semigroup of quotients of S. Let {Q_{i} : i ∈ I} be the set of distinct semigroups of quotients of S and let a^{∗(i)} denote the inverse of a ∈ S(S) in a subgroup of Q_{i}.

Put P = Π{Q_{i} : i ∈ I} and define ϕ : S → P by sϕ = (s), so that ϕ is an embedding of S into P. If a ∈ S(S) then a^{∗(i)} exists for all i ∈ I. Clearly (a) lies in a subgroup of P with inverse (a^{∗(i)}). Let

Q = ⟨(a)^{∗}, (b) : a ∈ S(S), b ∈ S⟩,

so that ϕ is an embedding of S into Q. It is easy to see that

Q = {(a)^{∗}(b) : a ∈ S(S), b ∈ S}

and it follows that S is an order in Q. Now if π_{i} : P → Q_{i} is the ith projection, π_{i} : Q → Q_{i} is an onto homomorphism and for any s ∈ S, sϕπ_{i} = (s)π_{i} = s, so that π_{i} is an S-homomorphism. Thus Q is the maximum semigroup of quotients of S.

Corollary 5.5 Let S be a commutative order. If S is a monoid, or if S = S(S), then S has a maximum semigroup of quotients.

6 Semilattice decompositions of commutative orders

A useful tool in studying semigroups is to break them down into smaller, hopefully simpler, constituent parts. One way of doing this is to consider semilattice decompositions of a semigroup. This philosophy has proved particularly useful in the study of left orders. For example, the question of whether a semigroup S is a left order in a regular semigroup with central idempotents (a Clifford semigroup) can be reduced to the study of semilattice congruences on S and the question of when a semigroup T is a left order in a group. This latter question is answered in [CP, Theorem 1.24]; T is a left order in a group if and only if T is right reversible and cancellative. Now if S is a left order in a Clifford semigroup Q, then as is well known, Q is a semilattice Y of groups G_{α}, α ∈ Y, and it is not difficult to see that S_{α} = S ∩ G_{α} is an order in G_{α} for each α ∈ Y. Thus each S_{α} is a right reversible cancellative semigroup and S is a semilattice Y of the semigroups S_{α}, α ∈ Y. On the other hand, given a semigroup T that is a semilattice Z of right reversible cancellative semigroups T_{γ}, γ ∈ Z, then T is an order in a Clifford semigroup P, where P is a semilattice Z of the groups of left quotients of the semigroups T_{γ}, γ ∈ Z. This is implicit in Theorem 3.1 of [4]. A similar approach is used in [9] to study straight left orders in completely regular semigroups. With this in mind we ask the following question, answered in the proposition below: if a commutative semigroup S is a semilattice Y of (commutative) semigroups S_{α}, α ∈ Y, and each S_{α} is an order in Q_{α}, when is S an order in a semigroup that is the union of the Q_{α}s?

Proposition 6.1 Let S be a commutative semigroup. Suppose that S is a semilattice Y of (commutative) semigroups S_{α}, α ∈ Y, where each S_{α} is an order in Q_{α}. Put

Q = ⋃{Q_{α} : α ∈ Y }
H′_{α} = H^{Q_{α}} ∩ (S_{α} × S_{α}), (α ∈ Y)

and

H′ = ⋃{H′_{α} : α ∈ Y }

so that H′ is an equivalence relation on S. Then Q is a semigroup of left quotients of S (under a multiplication extending that of each Q_{α}) if and only if H′ is a congruence on S and S(S) = ⋃{S(S_{α}) : α ∈ Y }. If these conditions hold then in addition Q is a semilattice Y of the semigroups Q_{α}, α ∈ Y.

Proof Suppose first that S is an order in Q. It is clear from Lemma 2.2 that Q is a semilattice Y of the semigroups Q_{α}, α ∈ Y. Now from the fact that for any α ∈ Y and any q ∈ Q_{α} there is an idempotent e ∈ Q_{α} with q = eq, it is easy to see that if p, q ∈ Q, then pHq in Q if and only if p, q ∈ Q_{α} and pHq in Q_{α}, for some α ∈ Y. It then follows that H′ is a congruence on S. Clearly S(S) ⊆ ⋃{S(S_{α}) : α ∈ Y } and the opposite inclusion also holds, since Q is a union of the quotient semigroups Q_{α}, α ∈ Y.

Conversely, suppose that H′ is a congruence on S and S(S) = ⋃{S(S_{α}) : α ∈ Y }.

Define a relation ≤ on S by the rule that for all b, c ∈ S,

b ≤ c if and only if bH′cd

for some d ∈ S. Note that if b ∈ S_{α} and c ∈ S_{β} then b ≤ c implies that α ≤ β.

For any α ∈ Y we write ≤_{α} for the relation ≤_{H^{Qα}} ∩(S × S). Then it is straightforward to show that for b, c ∈ S_{α},

b ≤ c if and only if b ≤_{α} c.

It is now easy to check that ≤ is a preorder compatible with multiplication, with associated equivalence relation H′. Conditions (A) and (C′) are immediately verified. To show that (B) holds, let b, c ∈ S and a ∈ S(S) with b ≤ a, c ≤ a and ab = ac. It follows that b, c ∈ S_{α} and a ∈ S_{β} for some α, β ∈ Y with α ≤ β. Choose x ∈ S(S_{α}) with b ≤_{α} x, so that b ≤ x and x ∈ S(S). Certainly every element of S(S) is H′-related to its square, so if bH′au, u ∈ S, then abH′a^{2}uH′auH′b and b ≤ ab. Further, bH′ab = acH′c, and b ≤ x gives xbH′b. Now xa ∈ S(S_{α}) and xabH′xbH′b, so that xab = xac and cH′_{α}b ≤_{α} xa yield b = c as required.

By Theorem 4.3, S is an order in a semigroup P where ≤_{H} ∩ (S × S) =≤. It is routine to check that P is a semilattice of semigroups P_{α} = {a^{∗}b : a, b ∈ S_{α}}, α ∈ Y and S_{α} is an order in P_{α}, α ∈ Y. As remarked in the first part of the proof, if p, q ∈ P then pHq in P if and only if p, q ∈ P_{α} and pHq in P_{α}, for some α ∈ Y. Now if u, v ∈ S_{α} then uHv in Q_{α} if and only if uH′v; but this is equivalent to uHv in P. Thus uHv in Q_{α} if and only if uHv in P_{α}. By Corollary 5.2, P_{α} is isomorphic over S_{α} to Q_{α}, and the proposition follows.

Corollary 6.2 [4] Let S be a commutative semigroup and suppose that S is a semilattice Y of cancellative semigroups S_{α}, α ∈ Y. Then S is an order in a commutative regular semigroup.

Proof In fact we show the stronger statement that S is an order in Q, where Q is a semilattice Y of the groups of quotients G_{α} of S_{α}, α ∈ Y.

Let Q and H′ be defined as in Proposition 6.1, where Q_{α} is replaced by G_{α}, α ∈ Y. Since H is the universal congruence on a group,

H′ = ⋃{S_{α} × S_{α} : α ∈ Y },

so that H′ is the congruence associated with the semilattice decomposition of S. Clearly S = ⋃{S(S_{α}) : α ∈ Y }. Let a, x, y ∈ S where a ∈ S_{α}, x ∈ S_{β} and y ∈ S_{γ} and suppose first that xa^{2} = ya^{2}. Then αβ = αγ and xa, ya ∈ S_{αβ}. Now cancelling xa in (xa)(ax) = (ya)(ax) we have xa = ya. On the other hand if xa^{2} = a^{2} then xa ∈ S_{α} and cancelling a gives xa = a. Thus a ∈ S(S) and so S = S(S) = ⋃{S(S_{α}) : α ∈ Y }.

¿From Proposition 6.1, S is an order in Q and Q is a semilattice Y of the groups G_{α}, α ∈ Y.

We note that in the above corollary the hypotheses may be weakened slightly: if S is a semilattice of commutative cancellative semigroups then it is not difficult to see that S itself is commutative. The converse of the corollary is also true, as shown in [4].

We also comment that if a commutative semigroup S is an order in Q and Q is a semilattice Y of semigroups Q_{α}, α ∈ Y, then it is not always the case that S_{α} = S ∩ Q_{α} is an order in Q_{α} for each α ∈ Y. This is illustrated by Example 7.1, where X^{G} is a chain of the group G and the null semigroup X. Now X^{T} ∩ X = X, but as 0 is the only element of X lying in a subgroup, certainly X is not an order in itself.

7 Examples

A left order S in Q is straight if any q ∈ Q can be expressed as q = a^{∗}b where a, b ∈ S and aRb in Q. As commented at the beginning of Section 4, a description of straight left orders is known [5]. It is easy to see that a commutative order S is straight (in any semigroup of quotients) if and only if S = S(S). With this in mind we comment that each of the orders presented in this section possesses elements that are not square-cancellable, and thus cannot be straight.

We show in Corollary 4.4 that if a commutative order S has the property that S(S) = S, then S is an order if and only if the preorder ≤_{H^{∗}} satisfies conditions (A), (B) and (C′). Our first example shows that if S ≠ S(S) then S can be an order without ≤_{H^{∗}} satisfying condition (C′). In spite of the fact that S ≠ S(S), S still has the property that its semigroups of quotients form a complete lattice.

If S is any semigroup with zero then we denote by S^{∗} the set of non-zero elements of S.

Example 7.1 .

Let R be any commutative semigroup and let X be a null semigroup disjoint from R. Then R ∪ X is a commutative semigroup under a multiplication extending that in R and X where

rx = x = xr

for all r ∈ R and x ∈ X. We write this semigroup as X^{R}.

Let T be a commutative cancellative semigroup and let X be a null semigroup disjoint from T with |X| ≥ 3. We consider the semigroup X^{T}.

It is easy to see that S(X^{T}) = {0} ∪ T and if T is an order in Q then X^{T} is an order in X^{Q}. Conversely, suppose that X^{T} is an order in P. Using the notation of Lemma 2.2, S(X^{T}) is an order in H(P). Write H(P) = {0} ∪ Q, so that {0} ∪ T is an order in {0} ∪ Q. Let p, q ∈ Q, so that pHa and qHb for some a, b ∈ T. Then pqHab so that pq ≠ 0 and Q is a subsemigroup, indeed a semigroup of quotients of T. If a ∈ T and x ∈ X then from ax = x we have x≤_{H}a in P, so that x = a^{∗}ax = a^{∗}x. It follows that P = X^{Q}.

Let Q = {P : X^{T} is an order in P } so that from the above comments,

Q = {X^{Q} : T is an order in Q}.

Define a relation ⪯ on Q by X^{Q} ⪯ X^{Q′} if there is an X^{T} -homomorphism from Q′ onto Q. At this point we note that if x, y ∈ X^{∗} and xHy in a semigroup of quotients X^{Q} of X^{T}, then from x = ya^{∗}b for some a, b ∈ X^{T} it follows that x = y. With this in mind it is easy to see from Theorem 5.1 that X^{Q} ⪯ X^{Q′} if and only if there is a T -homomorphism from Q′ onto Q. Proposition 5.3 now gives that Q is a complete lattice under ⪯. In particular, X^{T} has maximum and minimum semigroups of quotients.

T

X^{∗}

0 Fig. 7.1

Fig. 7.1 is the Hasse diagram of the preorder ≤_{H^{∗}} on X^{T}. Conditions (A), (B) and (C) hold for ≤_{H^{∗}}, but (C′) fails. For if b, c are distinct elements of X^{∗} then b≤_{H^{∗}}c, but bx ≠ cy for any x ∈ S(X^{T}), y ∈ X^{T} with b≤_{H^{∗}}x.

It has been independently conjectured by P.N. Anh that a commutative semigroup is an order if and only if ≤_{H^{∗}} satisfies conditions (A), (B) and (C). Our next example is a counterexample to this conjecture.

Example 7.2 .

Let R be a commutative semigroup and let X be a null semigroup disjoint from R. Then R ∪ X is a commutative semigroup under a multiplication extending that in R and X where

rx = 0 = xr

for all r ∈ R and x ∈ X. We write this semigroup as R_{X}.

Let T be a commutative cancellative semigroup with no idempotent and let X be a null semigroup disjoint from T with |X| ≥ 3. We consider the semigroup T_{X}^{1}.

It is straightforward to verify that S(T_{X}^{1}) = {1} ∪ T ∪ {0} and that if T is an order in Q then T_{X}^{1} is an order in Q_{X}^{1}. Conversely, suppose that T_{X}^{1} is an order in P. Write H(P) as H(P) = {1} ∪ Q ∪ {0}; as in the previous example, {1} ∪ T is an order in {1} ∪ Q. Suppose that p ∈ {1} ∪ Q and pH1 in P. Then p = a^{∗}b where a, b ∈ {1} ∪ T and bHp in P. Let x ∈ X^{∗}; then bxH1x = x so that bx ≠ 0 and b = 1. Now as pHa we must also have that ax ≠ 0 and so a = 1, giving also p = 1. In particular, if p, q ∈ H(P) and pq = 1, then p = q = 1. It follows that Q is a subsemigroup, T is an order in Q and further, P = Q_{X}^{1}. A similar argument to that in Example 7.1 shows that the set of semigroups of quotients of T_{X}^{1} is a complete lattice under ⪯, where P ⪯ P′ if and only if there is a T_{X}^{1}-homomorphism from P′ onto P.

Condition (B) fails for the relation ≤_{H^{∗}} on T_{X}^{1}: if a ∈ T and b ∈ X^{∗} then b≤_{H^{∗}}a but ab = 0 so b ≰_{H^{∗}}ab. Moreover, if b, c are distinct elements of X^{∗} then ab = ac = 0.

We now present examples of orders which do not possess a lattice of semigroups of quotients. The first is an order which has a maximum semigroup of quotients but not a minumum, the second is an order which has a minimum semigroup of quotients but not a maximum. The following lemma is useful in verifying that the multiplication we define in our examples is associative.

Lemma 7.3 Let X be a null semigroup and let T be a commutative semigroup disjoint from X which acts on X such that 0·t = 0 for all t ∈ T. Then S = X ∪T is a commutative semigroup under a multiplication extending that in X and T by

tx = xt = x · t

for all t ∈ T and x ∈ X.

Let T = {a_{i} : i ∈ Z^{+}} be the infinite monogenic semigroup generated by a and let

X = {x, y, 0} ∪ {z_{i} : i ∈ Z^{+}}

be a null semigroup disjoint from T. Define an action of T on X by

xa^{i} = ya^{i} = z_{i}, z_{j}a^{i} = z_{j+i}, 0a^{i} = 0

for all i, j ∈ Z^{+}. Let S = X ∪T be the semigroup with multiplication induced by this action; we consider the semigroup S^{1}. It is straightforward to check that S(S^{1}) = T ∪ {1, 0}.

Let ≤ be the preorder on S^{1} given by the Hasse diagram

{1}

T {y}

{x} ∪ {z_{i} : i ∈ Z^{+}}

{0}

It is routine to show that ≤ is compatible with multiplication and satisfies conditions (A), (B) and (C′). By Theorem 4.3, S^{1} is an order in a commutative semigroup Q where ≤_{H} ∩ (S × S) =≤. Thus in Q, x≤_{H}a.

Dually, S^{1} is an order in a commutative semigroup Q′ where in Q′ we have y≤_{H}a. If S^{1} had a minumum semigroup of quotients P, then Theorem 5.1 says that in P we have both x≤_{H}a and y≤_{H}a. Calculating in P gives

x = a^{∗}ax = a^{∗}ay = y,

a contradiction. Thus no such P exists. However, from Corollary 5.5, S^{1} has a maximum semigroup of quotients.

Let T = {a^{i}b^{j} : i, j ∈ N, i + j ≥ 1} be a subsemigroup of the free abelian group generated by a and b, and let

X = {(i, j) : i, j ∈ N} ∪ {0}

be a null semigroup disjoint from T. Define an action of T on X by

(i, j)a^{k}b^{ℓ} = (i + k, j + ℓ), 0a^{k}b^{ℓ} = 0

for (i, j) ∈ X and a^{k}b^{ℓ} ∈ T. Let S = X ∪ T be the semigroup with multiplication induced by this action.

It is easy to see that S(S) = {0} ∪ T. Let ≤ be the preorder on S given by the Hasse diagram

⟨a⟩

T \ ⟨a⟩ X_{H}

X_{L}

{0}

where ⟨a⟩ = {a^{i} : i ≥ 1}, X_{H} = {(i, 0) : i ∈ N} and X_{L} = {(i, j) : i, j ∈ N, j ≥ 1} = X^{∗} \X_{H}. Then ≤ is compatible with multiplication and satisfies conditions (A), (B) and (C′). By Theorem 4.3, S is an order in a commutative semigroup Q such that ≤_{H} ∩ (S × S) =≤. If u ∈ S(S) then in Q,

(0, 0)≤_{H}u if and only if u ∈ ⟨a⟩.

Dually, S is a left order in Q′ where for u ∈ S(S),

(0, 0)≤_{H}u if and only if u ∈ ⟨b⟩.

Suppose that S is an order in P where there exist S-homomorphisms from P onto Q and from P onto Q′. In P we must have (0, 0)≤_{H}u for some u ∈ S(S). By Theorem 5.1, (0, 0)≤_{H}u in Q and (0, 0)≤_{H}u in Q′. Thus u ∈ ⟨a⟩∩⟨b⟩ = ∅, a contradiction. Thus no such P exists and S has no maximum semigroup of quotients.

The preorder ≤_{H^{∗}} on S has Hasse diagram

T

X^{∗}

{0}

Certainly ≤_{H^{∗}} is compatible with multiplication, and it also satisfies conditions (A), (B) and (C′). Thus S is an order in a commutative semigroup R, where ≤_{H} ∩ (S × S) = ≤_{H^{∗}}. By the nature of ≤_{H^{∗}}, R is the minimum semigroup of quotients of S.

References

[1] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups Vol. I, American Math. Soc. 1961.

[2] J.B. Fountain,‘Abundant semigroups’, Proc. London Math. Soc. 44 (1982), 103-129.

[3] J.B. Fountain and M. Petrich, ‘Completely 0-simple semigroups of quotients’, J. Algebra 101 (1986), 365-402.

[4] V.A.R. Gould, ‘Clifford semigroups of left quotients’, Glasgow Math. J. 28 (1986), 181-191.

[5] V.A.R. Gould, ‘Straight left orders’, St. Sci. Math. Hungarica, to appear.

[6] J.M. Howie, An introduction to semigroup theory, Academic Press 1976.

[7] N.M. Khan, ‘On saturated permutative varieties and consequences of permutation identities’, J. Australian Math. Soc. 38 (1985), 186-197.

[8] D.B. McAlister, ‘One-to-one partial right translations of a right cancellative semigroup’, J. Algebra 43 (1976), 231-251.

[9] P.M. Smith, ‘Orders in completely regular semigroups’, D. Phil. thesis, University of York, 1990.

[10] M. Yamada, ‘Regular semigroups whose idempotents satisfy permutation identities’, Pacific J. Maths. 21 (1967), 371-392.