SIMPLIFIED CONDITIONS FOR TYPE-I AND TYPE-II MONOIDS CHRISTOPHER HOLLINGS Abstract. In a previous paper, we obtained conditions on a monoid M for its prefix expansion to be either a left restriction monoid (in which case M must be either ‘type-I’ or ‘type-II’) or a left ample monoid (M is ‘type-Ia’ or ‘typeIIa’). In the present paper, we demonstrate that there is some redundancy in these conditions. We therefore trim down the sets of conditions and show, by construction of suitable counterexamples, that the reduced sets of conditions are independent.

Introduction The notion of a semigroup expansion, a particular type of functor from one category of semigroups to a larger one [BR1], has proved to be an extremely useful tool in the study of the structure of semigroups — in particular, that of various classes of non-regular semigroups. Amongst these are left restriction semigroups, which arise in a very natural way from semigroups of partial transformations and thereby generalise inverse semigroups, and the special case of left ample semigroups, which arise from systems of injective partial transformations and form a class of semigroups intermediate between left restriction semigroups and inverse semigroups (for further details and references, see [Hol2]). In a previous paper [Hol1], we obtained necessary and sufficient conditions on a monoid for its so-called prefix expansion to be (i) a left restriction monoid,1 and (ii) a left ample monoid. In each instance, we found that the monoid must satisfy one of two sets of conditions; type-I and type-II were the names given to those monoids satisfying the conditions necessary for case (i) to hold, whilst type-Ia and type-IIa were the terms used in case (ii). Date: January 14, 2010. 2000 Mathematics Subject Classification. 20 M 99. Key words and phrases. monoid, unipotent, bipotent, type-I, type-II, type-Ia, type-IIa. ´ This work was carried out at the Centro de Algebra da Universidade de Lisboa (CAUL), supported by the Funda¸c˜ ao para a Ciˆencia e a Tecnologia (FCT) through the project ISFL1-143, as well as through FCT post-doctoral research grant SFRH/BPD/34698/2007. Thanks must go to James Woodward for help with counterexamples and also to various people at the International Conference on Semigroups and Related Topics (Porto, July 2009) for useful discussions. Some of the counterexamples presented here were constructed with the aid of the program Mace4 [PM]. 1 NB. In [Hol1], a left restriction monoid was termed a ‘weakly left E-ample monoid’. 1

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In the present paper, we revisit the results of [Hol1] and show that there is some redundancy in the conditions there obtained. The structure of the paper is as follows. We begin (Section 1) by recording the conditions for types-I, -Ia, -II, and -IIa monoids from [Hol1]. In Section 2, we examine the type-I and type-Ia conditions; in the former case, we prove, by the construction of suitable counterexamples, that the conditions given in Section 1 are already independent, whilst in the latter, we show that some simplification is possible. After carrying out this simplification, we prove that the remaining conditions are independent. Finally, in Section 3, we carry out a similar procedure for the type-II and type-IIa conditions; some simplification is possible in each case. In the interests of brevity, we omit the words ‘in a monoid M ...’ from the various results in this paper. Thus, for example, Theorem 2.1 reads simply ‘(U0) and (U1) are independent’, rather than the more precise ‘In a monoid M , (U0) and (U1) are independent’. For any unexplained semigroup-theoretic notation or terminology, the reader is referred to [How]. 1. Definitions Definition 1.1. A monoid is termed unipotent if it contains precisely one idempotent (necessarily its identity), and bipotent if it contains precisely two idempotents. The identity of an arbitrary monoid will always be denoted by 1; the second (non-identity) idempotent of an arbitrary bipotent monoid will always be denoted by e. Throughout this paper, G will denote the group of units of a monoid M ; the non-invertible elements of M form an ideal which we will denote by I. Thus · M = G ∪ I. Definition 1.2. A monoid M is termed a type-I monoid if it satisfies the following conditions: (U0) M is unipotent; (U1) su = s, for all s ∈ I and all u ∈ G. A type-I monoid M is termed a type-Ia monoid if it satisfies the following additional conditions: (U2) R = ι in I; (U3) u R v and us = vs ⇒ u = v; (U4) u R us = vs ⇒ u = v, where ι denotes the equality relation. (Note that we have phrased (U1) here in terms of G, unlike in [Hol1], where it is phrased in terms of H1 .) Definition 1.3. A monoid M is termed a type-II monoid if it satisfies the following conditions:

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(B0) M is bipotent; (B1) (a) R1 = {1}; (b) Re = {e}; (B2) e
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2.2. Type-Ia monoids. We begin with the following simplification: Lemma 2.2. (U4) ⇒ (U0). Proof. Suppose that M is a monoid in which (U4) holds and let x2 = x in M . We have x R xx = 1x. Then x = 1, by (U4).  Thus conditions (U1)–(U4) suffice to define a type-I monoid. In fact there is a further simplification which we may make, for which we must record a new condition on a monoid M = G ∪ I: (*U3) for u, v ∈ G and s ∈ I, us = vs ⇒ u = v. Lemma 2.3. (U2) and (*U3) ⇒ (U3). Proof. Suppose that M is a monoid in which (U2) and (*U3) hold and let u, v, s ∈ M be such that u R v and us = vs. If s is a unit, then it is immediate that u = v, so suppose that s ∈ I. There are two cases to consider: (a) u ∈ G. Since u R v, it follows from (U2) that v is also a unit and the result follows from (*U3). (b) u ∈ I. Then u = v, by (U2).  Consequently, conditions (U1), (U2), (*U3) and (U4) suffice to define a type-Ia monoid. Theorem 2.4. (U1), (U2), (*U3) and (U4) are independent. Proof. We present four counterexamples. In light of Lemma 2.2, each of the first three counterexamples that follow must be unipotent. Moreover, these three examples must be infinite, since a finite unipotent monoid is necessarily a group, but all of our conditions hold in a group. (U2), (*U3) and (U4) 6⇒ (U1). Let Z∗ be the unipotent monoid of all nonzero integers under multiplication. We know from the proof of Theorem 2.1 that (U1) fails in Z∗ . It remains to check that (U2), (*U3) and (U4) hold. It is easily verified that all R-classes in I = {. . . , −3, −2, 2, 3, . . .} are singletons. Thus (U2) holds. Moreover, (*U3) and (U4) follow immediately from right cancellation in Z∗ . (U1), (*U3) and (U4) 6⇒ (U2). Let X be a countably infinite set, and let BL(X) be the Baer-Levi semigroup on X: the semigroup of all one-one mappings α : X → X (with left-to-right composition) such that X \ Xα is infinite. Then BL(X) is right cancellative and idempotent-free, and R is universal in BL(X) [CP, Theorem 8.2]. We adjoin an identity 1 to BL(X) to obtain a unipotent monoid BL(X)1 . Then 1 is the only unit in BL(X)1 , so (U1) holds. Moreover, (*U3) and (U4) follow from right cancellation. However, (U2) fails, since R is universal in I = BL(X). (U1), (U2) and (U4) 6⇒ (*U3). Let hai be the infinite cyclic group on a generator a and let hbi be the infinite monogenic semigroup on a generator b. We put M = hai ∪ hbi and define multiplication in M by putting ai bj = bj = bj ai , for

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i ∈ Z and j ∈ N; multiplication of two elements from hai, or of two elements from hbi, is simply that in the original group or semigroup, respectively. It is easily verified that this multiplication is associative, and that the only idempotent in M is a0 , the identity in hai, which is also an identity for M . Thus M is a unipotent monoid with G = hai. Condition (U1) holds in M (su = bj ai = bj = s), as does (U2), since R = ι in hbi = I. As for (U4), we observe that the left-hand side of the implication can only hold when s ∈ G. To see this, suppose that s ∈ I, so that s = bi , for some i ∈ N. Then, in order to have u R us, we must have u = usx, for some x ∈ M . It follows that u ∈ I, since s ∈ I and I is an ideal. Thus u = bj , for some j ∈ N. But then u = usx gives ( bj+i if x ∈ G; bj = bj+i x = j+i+k b if x = bk ∈ I. In either case, we have a contradiction, since i, j, k > 0. Thus, in order for the left-hand side of (U4) to hold, we must have s ∈ G, in which case the right-hand side follows immediately, and so (U4) holds. We must finally demonstrate that (*U3) does not hold in M ; we do so by noting that ai bj = ak bj in M , even if i 6= k. (U1), (U2) and (*U3) 6⇒ (U4). Let C2 be the 2-element chain, as in the proof of Theorem 2.1. We know that (U1) holds. Also, since C2 is a semilattice, we have R = ι, so (U2) holds. Moreover, (*U3) holds, because G = {1}. However, (U4) does not hold, since 0 R 0 · 0 = 1 · 0.  3. Types-II and -IIa monoids In this section, we consider types-II and -IIa monoids; some simplification of conditions is possible in each case. 3.1. Type-II monoids. We begin with the following simplifications: Lemma 3.1. (B0) and (B3) ⇒ (B1)(a). Proof. Let M be a monoid in which (B0) and (B3) hold and suppose that x R 1 in M . Then there exists y ∈ M such that xy = 1. If y = 1, then it follows that x = 1 and we are done. On the other hand, suppose that y 6= 1. Then e is a right identity for y and we have xye = xy = e. But this gives e = 1, a contradiction. Therefore y = 1, whence x = 1.  Lemma 3.2. (B0), (B1)(b) and (B3) ⇒ (B2). Proof. Let M be a monoid in which (B0), (B1)(b) and (B3) hold and suppose that e
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Thus, conditions (B0), (B1)(b) and (B3) suffice to define a type-II monoid. Theorem 3.3. (B0), (B1)(b) and (B3) are independent. Proof. We present three counterexamples. (B1)(b) and (B3) 6⇒ (B0). Let M3.3(a) be the monoid with the following multiplication table: 1 1 1 e e f f

e e e f

f f e f

Note that 1M3.3(a) = M3.3(a) , eM3.3(a) = {e} and f M3.3(a) = {f }, so R = ι in M3.3(a) , in which case, (B1)(b) holds. Furthermore, it is easy to see from the table that (B3) holds also. Note, however, that (B0) fails, since f is idempotent. (B0) and (B3) 6⇒ (B1)(b). Let M3.3(b) be the monoid with the following multiplication table: 1 e s 1 1 e s e e e s s s s e Then conditions (B0) and (B3) are clear. Note, however, that 1M3.3(b) = M3.3(b) and eM3.3(b) = {e, s} = sM3.3(b) , so Re = {e, s} and (B1)(b) fails. (B0) and (B1)(b) 6⇒ (B3). Let M3.3(c) be the monoid with the following multiplication table: 1 1 1 s s 0 0

s s 0 0

0 0 0 0

Then M3.3(c) is certainly bipotent, with idempotents 1 and 0. Moreover, 1M3.3(c) = M3.3(c) , sM3.3(c) = {s, 0} and 0M3.3(c) = {0}, so R = ι and (B1)(b) holds. However, 0 is clearly not a right identity for s, so (B3) does not hold.  3.2. Type-IIa monoids. We begin by noting a simplification to the type-IIa conditions which follows from the preceding subsection. Since (B1)(b) follows from (*B1), we see that a type-IIa monoid satisfies conditions (B0), (B1)(b) and (B3). But (B2) follows from these, by Lemma 3.2. Conditions (B0), (*B1), (B3) and (B4) are therefore sufficient to define a type-IIa monoid. We now introduce a new condition on a monoid M , a weakening of (B0): (*B0) M contains an idempotent e 6= 1. Lemma 3.4. (*B0) and (B4) ⇒ (B0).

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Proof. Let M be a monoid in which conditions (*B0) and (B4) hold. Suppose that a2 = a but a = 6 e. We put b = a and c = 1 in (B4) and conclude that a = b = 1.  We see then that (*B0), (*B1), (B3) and (B4) are sufficient to define a type-IIa monoid. Theorem 3.5. (*B0), (*B1), (B3) and (B4) are independent. Proof. We present four counterexamples. (*B1), (B3) and (B4) 6⇒ (*B0). Note that the counterexample in this instance must be unipotent and that (B3) therefore becomes vacuous, since there is no such e. Let hαi1 be the infinite monogenic semigroup on a generator α, with an identity 1 adjoined. This is certainly a unipotent monoid. Observe that hαi is R-trivial and that this property carries over into hαi1 , so (*B1) holds. Finally, ignoring the condition ‘a 6= e’, we see that (B4) follows from cancellation in hαi1 . (*B0), (B3) and (B4) 6⇒ (*B1). Let G be any group and denote the identity of G by e. We adjoin a second identity to G to obtain a bipotent monoid G1 . It is clear that (B3) holds in G1 , but that (*B1) does not, since the R-classes of G1 are G and {1}. Condition (B4) follows from cancellation in G whenever a, b, c 6= 1. The other cases are easily dealt with: if a = 1, then it is immediate that b = c; if b = 1, then a = 1 = c also; if c = 1, then a = b, whence b2 = b, and so b = 1 = c, since G1 is bipotent and b = a 6= e. (*B0), (*B1) and (B4) 6⇒ (B3). Let hαi be the infinite monogenic semigroup on α, as above. We adjoin first a zero 0 and then an identity 1 to hαi to obtain a bipotent monoid hαi0,1 . As in the first part of the proof, the fact that hαi is R-trivial carries over to hαi0,1 and so (*B1) holds. Since hαi is cancellative, we have that ba = b = ca implies b = c in hαi0,1 , provided a 6= 0, which is precisely condition (B4). Finally, note that (B3) does not hold, as e = 0 is not a right identity for hαi0 . (*B0), (*B1) and (B3) 6⇒ (B4). We consider the monoid M3.3(a) in the proof of Theorem 3.3. We already know from the proof of that theorem that both (*B1) and (B3) hold in M3.3(a) ; (*B0) is clear. Note however that (B4) does not hold, since f 2 = f = 1f .  Notice that the first and third counterexamples in the proof of this theorem are infinite, whilst the fourth is a small finite example; the second may be either finite or infinite. The first counterexample is necessarily infinite since, in order to negate (*B0), we must assume that the monoid is unipotent; a finite unipotent monoid is necessarily a group, and this would not satisfy (*B1). The following lemma,2 which should be compared with [Hol1, Proposition 6.1], shows that the third counterexample must be infinite as well. 2Thanks

must go to the anonymous referee for supplying this result.

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Lemma 3.6. A monoid which satisfies conditions (*B0), (*B1) and (B4) is either infinite or isomorphic to the 2-element chain C2 . Proof. Suppose that M is a finite monoid which satisfies conditions (*B0), (*B1) and (B4), and suppose that M has a non-idempotent element s. Then s has an idempotent power, which we denote by sω . We let sω−1 be the power of s which satisfies sω−1 s = sω . Observe that sω = (ssω )sω−1 and ssω = sω s, from which it follows that sω R ssω , and so sω = ssω , by (*B1). However, we may then set a = s, b = sω and c = sω−1 in (B4) to obtain a contradiction. We conclude therefore that M is a band. Now, if |M | > 2, then M contains an element b other than 1 and e. But we may then put a = b and c = 1 in (B4) to obtain another contradiction, from which we conclude that |M | = 2. Thus, M ∼  = C2 . We conclude by presenting the following table, which summarises the simplifications carried out for the various classes of monoids considered in this paper: Monoid Type-I Type-Ia Type-II Type-IIa

Independent defining conditions (U0) and (U1) (U1), (U2), (*U3) and (U4) (B0), (B1)(b) and (B3) (*B0), (*B1), (B3) and (B4) References

[BR1] J.-C. Birget and J. Rhodes, Almost finite expansions of arbitrary semigroups, J. Pure Appl. Algebra, 32 (1984), 239-287. [CP] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Volume 2, Mathematical Surveys of the Amer. Math. Soc., No. 7, Amer. Math. Soc., Providence, R. I., 1967. [Hol1] C. Hollings, Conditions for the prefix expansion of a monoid to be (weakly) left ample, Acta Sci. Math. (Szeged), 73(3–4) (2007), 519–545. [Hol2] C. Hollings, From right PP monoids to restriction semigroups: a survey, Europ. J. Pure Appl. Math., 2(1) (2009), 21–37. [How] J. M. Howie, Fundamentals of Semigroup Theory, LMS Monographs No. 12, Clarendon Press, Oxford, 1995. [PM] Prover9 and Mace4, http://www.cs.unm.edu/∼mccune/prover9/ School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK E-mail address: [email protected]