The Rank of a Commutative Cancellative Semigroup

Acta Math. Hungar. 107 (1–2) (2005), 71–75.

THE RANK OF A COMMUTATIVE CANCELLATIVE SEMIGROUP

A. M. CEGARRA (Granada) and M. PETRICH (Zagreb)∗

Abstract. For a commutative cancellative semigroup S, we deﬁne the rank of S intrinsically. This deﬁnition implies that the rank of S equals the usual rank of its group of quotients. We also characterize the rank in terms of embeddability into a rational vector space of the greatest power cancellative image of S.

1. Introduction

It is natural to assign to a commutative cancellative semigroup S the rank of its group G of quotients. If this definition is adopted, then for an abelian group G, all orders in G have the same rank, namely that of G. While these statements are coherent, it would be desirable to have an intrinsic definition of the rank of a commutative cancellative semigroup. The rank of an abelian group G is defined as the cardinality of any maximal independent subset of G. Then the rank of G is the sum of torsion-free rank of G and p-ranks of G. We are concerned here with the torsion-free rank of G which is the cardinality of any maximal independent subset of G consisting of elements of infinite order. The first task is to find an analogue of independence for commutative cancellative semigroups. If this is done correctly, the definition should yield the properties which are appropriate for semigroups at the same time repre- senting analogues of statements true for an independent set in the context of groups. We also require coherence of the two concepts of rank, namely if S is an order in an abelian group G, then the rank of S should be the same as the rank of G. In addition, the semigroup definition should yield the group definition if the former is applied to groups. With all these requirements, there should be left but little choice for the definition of rank of a commutative cancellative semigroup.

∗The authors were partially supported by DGI of Spain and FEDER, Project: BFM2001-2886. Key words and phrases: semigroup, commutative, cancellative, group of quotients, rank. 2000 Mathematics Subject Classiﬁcation: 20M10, 20M14.

0236–5294/5/$ 20.00 c 2005 Akad´emiai Kiad´o,Budapest 72 A. M. CEGARRA and M. PETRICH

Some terminology and notation comprises Section 2. The rank of a commutative cancellative semigroup is introduced in Section 3 with the necessary justiﬁcation. Section 4 contains a characterization of the rank in terms of rational vector spaces.

2. Terminology and notation

For these we follow generally the books [1] and [2]. The second refer- ence contains extensive discussion of our subject. Throughout S denotes a commutative cancellative semigroup. We shall thus omit these qualiﬁers, in particular all groups will be abelian. We denote by P and Z the sets of positive and all integers, respectively. We let SQ = (S × S)/ ∼, where ∼ is deﬁned by

(a, b) ∼ (c, d) ⇐⇒ ad = bc with multiplication of ∼-classes [a, b][c, d] = [ac, bd]. Then SQ is the group of quotients of S. A subsemigroup S of a group G is an order in G if every element g ∈ G admits a representation g = ab−1 for some a, b ∈ S. In particular, the canonical injection δ : a 7→ [a2, a], a ∈ S, maps S onto an order in SQ.

3. Rank

A subset {a1, . . . , ak} of an abelian group G is said to be independent if n1 nk a1 ··· ak = e, where ni ∈ Z and e is the identity of G, implies that n1 = ··· = nk = 0. An inﬁnite subset of G is called independent if each of its ﬁnite subsets is independent. The (torsion-free) rank of G, denoted by rank G, is the cardinality of a maximal independent subset of G. According to ([1], Theorem 16.3), rank G is an invariant of G, that is all maximal independent subsets of G have the same cardinality. We are thus led naturally to the following fundamental concept.

Definition 3.1. A subset {a1, . . . , ak} of S is independent if

m1 mk n1 nk a1 ··· ak = a1 ··· ak for some mi, ni ∈ P implies mi = ni for i = 1, . . . , k. An inﬁnite subset of S is independent if all its ﬁnite subsets are independent. It follows that a nonempty subset L of S is independent if and only if the subsemigroup of S generated by L is freely generated by L as a commutative semigroup.

Acta Mathematica Hungarica 107, 2005 THE RANK OF A COMMUTATIVE CANCELLATIVE SEMIGROUP 73

If S is a subsemigroup of a group G, for example its group of quotients SQ, we may ask how the notions of independence in S and G compare. Since m1 mk n1 nk m1−n1 mk−nk a1 ··· ak = a1 ··· ak in S if and only if a1 ··· ak = e in G, the answer is very simple: L is independent in S if and only if it is independent in G. Proposition 3.2. Let S be an order in a group G and let L j S. Then L is a maximal independent subset of S if and only if L is a maximal independent subset of G. Proof. If L is maximal in S, then for any g = ab−1 ∈ G, where m1 mh m n1 nk n a, b ∈ S, there are relations a1 ··· ah a = e, b1 ··· bk b = e in G, with m1n mhn −n1m −nkm mn ai, bi ∈ L and m, n 6= 0, which imply a1 ··· ah b1 ··· bk g = e, showing that L is maximal in G. The following theorem will prepare the way for an intrinsic deﬁnition of rank of a commutative cancellative semigroup. Theorem 3.3. (i) Every independent subset of S can be completed to a maximal independent subset of S. (ii) All maximal independent subsets of S have the same cardinality. Proof. (i) This follows by a standard Zorn’s lemma argument. (ii) This is a consequence of Proposition 3.2 and the invariance of rank of an abelian group ([1], Theorem 16.3). Definition 3.4. The rank of S, denoted by rank S, is the cardinality of any maximal independent subset of S. Corollary 3.5. The rank of an abelian group equals the rank of any of its orders. In particular rank S = rank SQ. Corollary 3.5 amounts to an alternative (external) way of deﬁning the rank of a commutative cancellative semigroup adopted in ([3], Section 4) and in [4].

4. A characterization of rank

The dimension of a vector space V is at the same time the cardinality of any maximal linearly independent subset of V and of any minimal generator set of V . In the preceding section we deﬁned the rank of S in analogy with the ﬁrst of these characterizations; in this section we provide an analogue of the second. We will need some preliminaries. A semigroup S is power cancellative if for any a, b ∈ S and n ∈ P, an = bn implies that a = b. A congruence ρ on S is power cancellative if the quotient semigroup S/ρ is. As usual, congruences on S are ordered by inclusion as binary relations. We start with a simple but important auxiliary result.

Acta Mathematica Hungarica 107, 2005 74 A. M. CEGARRA and M. PETRICH

Lemma 4.1. On S deﬁne a relation τ by

x τ y if xn = yn for some n ∈ P.

Then τ is the least power cancellative congruence on S. In the sequel we shall use the notation τ as above. Lemma 4.2. rank S = rank S/τ. Proof. The congruence τ is induced by the torsion part T (SQ) of SQ, so that S/τ is isomorphic to an order in SQ/ T(SQ) and (S/τ)Q =∼ SQ/ T(SQ). Now SQ and SQ/ T(SQ) have the same (torsion-free) rank; therefore, by Corollary 3.5, S and S/τ have the same rank. The rank of a free commutative semigroup F is usually deﬁned as the cardinality of a set of free generators of F . We then see that our rank coincides with this rank. Using this concept, we have that rank S equals the rank of a maximal free commutative subsemigroup of S. We shall now see that commutative cancellative power cancellative semigroups are precisely those embeddable into (the additive group of) rational vector spaces. Theorem 4.3. The semigroup S is power cancellative if and only if S is embeddable into a rational vector space V . In such a case, rank S coincides with the dimension of the subspace of V generated by the image of S in V . Proof. Necessity. It follows easily that power cancellativity in S implies torsion-freeness in SQ. By ([1], Theorems 23.1 and 24.4) the divisible hull SQE of SQ is a rational vector space with dim SQE = rank SQ. Since rank SQ = rank S by Corollary 3.5, we conclude that dim SQE = rank S. Let S j V , where V is a rational vector space. By the universal property of the group of quotients, we get a canonical embedding SQ j V which can be extended to an embedding SQE j V . It follows that the subspace of V generated by S is (isomorphic to) SQE, and thus is of dimension equal to rank S. Suﬃciency. The additive group of a rational vector space is a power cancellative semigroup. Since this property is inherited by subsemigroups, the assertion follows. We thus arrive at the desired characterization. Corollary 4.4. Let V be a rational vector space into which S/τ can be embedded. Then rank S is the cardinality of any minimal generating set of the subspace of V generated by the image of S/τ.

Acknowledgment. The authors are indebted to the referee for substan- tial improvement of the paper.

Acta Mathematica Hungarica 107, 2005 THE RANK OF A COMMUTATIVE CANCELLATIVE SEMIGROUP 75

References

[1] L. Fuchs, Inﬁnite Abelian Groups I, Academic Press (New York, San Francisco, Lon- don, 1970). [2] P. A. Grillet, Semigroups, An Introduction to Structure Theory, Dekker (New York, 1995). [3] H. B. Hamilton, T. E. Nordahl and T. Tamura, Commutative cancellative semigroups without idempotents, Paciﬁc J. Math., 61 (1975), 441–456. [4] Y. Kobayashi, Conditions for commutative semigroups to have nontrivial homomor- phisms into nonnegative (positive) reals, Proc. Amer. Math. Soc., 65 (1977), 199–203.

(Received July 29, 2003; revised September 29, 2003)

DEPARTAMENTO DE ALGEBRA´ FACULTAD DE CIENCIAS UNIVERSIDAD DE GRANADA 18071 GRANADA ESPANA˜ E-MAIL: [email protected]

DEPARTMENT OF MATHEMATICS UNIVERSITY OF ZAGREB 10002 ZAGREB CROATIA E-MAIL: [email protected]

Acta Mathematica Hungarica 107, 2005