Free Split Bands Francis Pastijn Marquette University, [email protected]

Free Split Bands Francis Pastijn Marquette University, Francis.Pastijn@Marquette.Edu

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by epublications@Marquette

Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of

6-1-2015 Free Split Bands Francis Pastijn Marquette University, [email protected]

Justin Albert Marquette University, [email protected]

Accepted version. Semigroup Forum, Vol. 90, No. 3 (June 2015): 753-762. DOI. © 2015 Springer International Publishing AG. Part of Springer Nature. Used with permission. Shareable Link. Provided by the Springer Nature SharedIt content-sharing initiative. NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

Free Split Bands

Francis Pastijn Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI Justin Albert Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI

Abstract: We solve the word problem for the free objects in the variety consisting of bands with a semilattice transversal. It follows that every free band can be embedded into a band with a semilattice transversal.

Keywords: Free band, Split band, Semilattice transversal 1 Introduction

We refer to3 and6 for a general background and as references to terminology used in this paper.

Recall that a band is a semigroup where every element is an idempotent. The Green relation is the least semilattice congruence on a band, and so every band is a semilattice of its -classes; the -classes themselves form rectangular bands.5 We shall be interested in bands𝒟𝒟 S for which the least semilattice congruence splits, that is, there exists a subsemilattice𝒟𝒟 of 𝒟𝒟 which intersects each -class in exactly one element. Such a subsemilattice ∘ of will then be called a semilattice𝒟𝒟 transversal of . 𝑆𝑆∘ 𝑆𝑆 𝒟𝒟 𝑆𝑆 𝑆𝑆 𝑆𝑆 Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 1

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

If the band has a semilattice transversal , then we can associate to every the unique which is -related to a. The unary ∘operation , is an idempotent - class preserving∘ ∘ endomorp𝑆𝑆 hism of which induces𝑆𝑆 the -relation on , and in∘ particular,𝑎𝑎 ∈ 𝑆𝑆 is a 𝑎𝑎 ∈ 𝑆𝑆 𝒟𝒟 𝑆𝑆 ↦ 𝑆𝑆 𝑎𝑎 ↦ 𝑎𝑎 𝒟𝒟 retract of . The unary band (( , , )) thus obtained obviously satisfies, apart from the associative∘ law, the identities ∘𝑆𝑆 𝒟𝒟 𝑆𝑆 𝑆𝑆 𝑆𝑆 𝑆𝑆 ⋅ , , ,

( ) 2 ∘ , (∘ )∘ ∘ . 𝑥𝑥 ≈ 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 ≈ 𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 ≈ 𝑥𝑥 (1) ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ 𝑥𝑥𝑥𝑥 ≈ 𝑥𝑥 𝑦𝑦 ≈ 𝑦𝑦 𝑥𝑥 𝑥𝑥 ≈ 𝑥𝑥 One readily verifies that the last identity ( ) follows in fact from the previous ones. It is not difficult to see that if a unary band (( , , )∘) satisfies∘ ∘ the identities (1), then = 𝑥𝑥 ≈ 𝑥𝑥 { | } is a semilattice transversal of the band∘ ( , ). For this reason we call the unary∘ bands which∘ satisfy the identities (1) split bands. 𝑆𝑆 ⋅ 𝑆𝑆 𝑎𝑎 𝑎𝑎 ∈ 𝑆𝑆 𝑆𝑆 ⋅ The variety of bands will be denoted by and the variety of all split bands will be denoted . For a nonempty set , ( ) and ( ) denote the free objects on in 𝐁𝐁 and , respectively.∘ As the abstract indicates, the∘ purpose of this paper is to solve the 𝐁𝐁 𝑋𝑋 𝐹𝐹𝐁𝐁 𝑋𝑋 𝐹𝐹𝐁𝐁 𝑋𝑋 𝑋𝑋 𝐁𝐁 word problem∘ for ( ) and to show that FB(X) can be isomorphically embedded into 𝐁𝐁 the multiplicative reduct∘ of ( ). The solution of the word problem for ( ) is well 𝐹𝐹𝐁𝐁 𝑋𝑋 understood and the reader will∘ find all the necessary details in Sect 4.5 of3 where 𝐁𝐁 𝐁𝐁 appropriate references to1 and𝐹𝐹 2 are𝑋𝑋 given. Our solution to the word problem𝐹𝐹 for𝑋𝑋 ( ) is similar though slightly more complicated. While is, like , locally finite, for any finite∘ 𝐁𝐁 nonempty set the (finite) free object ( ) is much∘ larger than ( ). 𝐹𝐹 𝑋𝑋 ∘ 𝐁𝐁 𝐁𝐁 𝐁𝐁 𝐁𝐁 There is𝑋𝑋 something more enigmatic𝐹𝐹 about𝑋𝑋 all this. It turns out𝐹𝐹 that𝑋𝑋 if a band has a semilattice transversal then the union of all the -classes [ -classes] in S of the elements of is a left [right]∘ regular band which is a transversal of the -classes [𝑆𝑆 - classes] of . This∘ follows𝑆𝑆 from a very special applicationℒ of Propositionℛ 2.3 and Corollary 2.4𝑆𝑆 of8 and below we intend to give a short independent proof ℛof this fact ℒin the special circumstances𝑆𝑆 we consider here.

Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 2

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

Result 1.1

Let ( , , ) be a split band with semilattice transversal = { | }. Then = { | } is∘ a left [right] regular subband of ( , ) with semilattice∘ transversal∘ , and 𝑆𝑆 is∘ a transversal𝑆𝑆 ⋅ of the -classes [ -classes] of ( , ). 𝑆𝑆 𝑎𝑎 𝑎𝑎 ∈ 𝑆𝑆 ∘ 𝐼𝐼 𝑎𝑎𝑎𝑎 𝑎𝑎 ∈ 𝑆𝑆 𝑆𝑆 ⋅ 𝑆𝑆 𝑆𝑆 PΛroof ℛ ℒ 𝑆𝑆 ⋅

Clearly, for any , the intersection of and the -class of is the the -class of and intersects the -class of in the unique element . It suffices to prove that is∘ 𝑎𝑎 ∈ 𝑆𝑆 𝐼𝐼𝑆𝑆 𝒟𝒟 𝐷𝐷𝑎𝑎 𝑎𝑎 ℒ 𝐿𝐿𝑎𝑎 a sub∘semigroup of , or in particular, that for any , we have that∘ ( )( ) ( ) . In 𝑎𝑎 𝐼𝐼𝑆𝑆 ℛ 𝑅𝑅𝑎𝑎 𝑎𝑎 𝑎𝑎𝑎𝑎 𝐼𝐼𝑆𝑆 any case = = ( ) is -related to ( )( ) in and since∘ this ∘-class is a∘ 𝑆𝑆 𝑎𝑎 𝑏𝑏 ∈ 𝑆𝑆 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 ℒ 𝑎𝑎𝑎𝑎 rectangular∘ band∘ it follows∘ ∘ that (∘ )( ) ( ∘ )( ∘ ). Applying the identities (1) we find that 𝑎𝑎 𝑏𝑏 𝑏𝑏 𝑎𝑎 𝑎𝑎𝑎𝑎 ∘ 𝒟𝒟∘ ∘ ∘ 𝑎𝑎𝑎𝑎 ∘𝑏𝑏𝑏𝑏 ∘ 𝑆𝑆 𝒟𝒟 𝑏𝑏 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 ℒ 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 ( )( ) = ( ) ∘ ∘ ∘ ∘ = ∘ ∘ ∘ ∘

𝑏𝑏 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 = 𝑏𝑏∘ 𝑎𝑎∘ 𝑎𝑎𝑎𝑎∘ 𝑏𝑏𝑏𝑏 = 𝑏𝑏∘𝑎𝑎∘𝑏𝑏=𝑏𝑏∘( ) . 𝑎𝑎∘𝑏𝑏∘𝑏𝑏𝑏𝑏 ∘ Thus ( )( ) ( ) as required. 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎 ∘ ∘ ∘ 𝑎𝑎The𝑎𝑎 𝑏𝑏 and𝑏𝑏 ℒ 𝑎𝑎 mentioned𝑎𝑎 in Result□ 1.1 may well serve as a means to coordinatize and one would expect that such a coordinatization would set the stage for a structure 𝑆𝑆 𝑆𝑆 theorem of 𝐼𝐼split bandsΛ in terms of the left and right regular bands and akin to, but 𝑆𝑆 simpler than the construction in II.1 of.6 For a free split band = ( ), it is easy to 𝐼𝐼𝑆𝑆 Λ𝑆𝑆 characterize the elements of the left [right] regular split band (see∘ Corollary 2.2 and 𝐁𝐁 Theorem 2.5). By left-right duality, is anti-isomorphic to 𝐹𝐹. 𝐹𝐹 𝑋𝑋 Λ𝐹𝐹 𝐹𝐹 𝐹𝐹 The variety of right𝐼𝐼 [left] regular split bandsΛ is the subvariety of determined by the additional∘ identity∘ . Thus, with the notation of∘ 𝐑𝐑𝐑𝐑𝐑𝐑 𝐋𝐋𝐋𝐋𝐋𝐋 𝐁𝐁 Result 1.1, and belong to and , respectively. As we shall see, if = 𝑥𝑥𝑥𝑥𝑥𝑥 ≈ 𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥𝑥𝑥 ≈ 𝑥𝑥𝑥𝑥 ( ) is a free split band, then and∘ should∘ not be assumed to be free on in 𝐼𝐼𝑆𝑆 Λ𝑆𝑆 𝐋𝐋𝐋𝐋𝐋𝐋 𝐑𝐑𝐑𝐑𝐑𝐑 𝐹𝐹 and∘ , respectively. ∘ 𝐁𝐁 𝑆𝑆 𝑆𝑆 𝐹𝐹 𝑋𝑋 ∘ 𝐼𝐼 Λ 𝑋𝑋 𝐋𝐋𝐋𝐋𝐋𝐋 𝐑𝐑𝐑𝐑It𝐑𝐑 is time to put our paper in the context of current research. The adequate terminology split band is not of our invention but already occurs in4 where the authors give

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

a structure theorem for orthodox semigroups for which the least inverse semigroup congruence splits. Theorem 2 of9 gives a structure theorem for the members of in the manner of Theorem II.1.6 of.6 Combining this result of Yoshida with its dual and with∘ Theorem 2 of,7 one obtains a structure theorem for the members of . We would,𝐑𝐑𝐑𝐑 𝐑𝐑 however, like to draw the reader’s attention to the all encompassing paper∘ 8 which has already been mentioned above, and which in its Example 2.15 introduces𝐁𝐁 a variety of unary semigroups (whose members are all regular semigroups) which contains as a subvariety. ∘ 𝐁𝐁 2 Free split bands

In this section we give a solution of the word problem for the free object ( ) in the variety on a nonempty set of variables. ∘ 𝐁𝐁 ∘ 𝐹𝐹 𝑋𝑋 We let𝐁𝐁 be a set disjoint of 𝑋𝑋 and , , a bijection. The elements of will be called letters∘ . The identity of the free monoid∘ ( ∘ ) is the empty word 1, thus ( ∘ 𝑋𝑋 𝑋𝑋 𝑋𝑋 ↦ 𝑋𝑋 𝑥𝑥 ↦ 𝑥𝑥 𝑋𝑋 ∪ 𝑋𝑋 ) = ( ) {1}. For any ( ) we define∘ ∗ the content ( ) of inductively∘ ∗ by ∘ + ∘ ∗𝑋𝑋 ∪ 𝑋𝑋 𝑋𝑋 ∪ 𝑋𝑋 𝑋𝑋 ∪ 𝑋𝑋 ∪ 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑐𝑐 𝑤𝑤 𝑤𝑤 (1) = , ( ) = ( ) = { }, ,

( 𝑐𝑐 ) = ∅( ∘ ) ( ), 𝑐𝑐 𝑥𝑥 >𝑐𝑐 1𝑥𝑥, , …𝑥𝑥, 𝑥𝑥 ∈ 𝑋𝑋. 1 𝑛𝑛−1 𝑛𝑛 1 𝑛𝑛−1 𝑛𝑛 𝑐𝑐 𝑦𝑦 ⋯ 𝑦𝑦 𝑦𝑦 𝑐𝑐 𝑦𝑦 ⋯ 𝑦𝑦 ∪ 𝑐𝑐 𝑦𝑦 ∘ We let be the congruence relation of𝑛𝑛 the free𝑦𝑦 semigroup1 𝑦𝑦𝑛𝑛 (∈ 𝑋𝑋 ∪)𝑋𝑋 generated by the pairs ∘ + 𝛽𝛽 𝑋𝑋 ∪ 𝑋𝑋 ( , ), ( ) , ( , 2), ( , ), ∘ + , 𝑤𝑤( 𝑤𝑤∘ , ∘ )𝑤𝑤,∘ ∈∘ 𝑋𝑋, ∪ 𝑋𝑋 . 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 (2) ∘ ∘ ∘ ∘ 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 One readily verifies that ( )/ is a band generated by the elements of the form or , and Green’s -relation on this band∘ is given by ∘ 𝑋𝑋 ∪ 𝑋𝑋 𝛽𝛽 𝑥𝑥𝑥𝑥 𝑥𝑥 𝛽𝛽 𝒟𝒟 ( ) = ( ).

(3) Semigroup forum, Vol. 90, No. 3 (June 2015): pg.𝑣𝑣𝑣𝑣 753𝒟𝒟-762𝑤𝑤. DOI𝑤𝑤. This⇔ article𝑐𝑐 is𝑣𝑣 © Springer𝑐𝑐 and𝑤𝑤 permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 4

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

Also, the element of the form ( … ) , 1, , … , , constitute a subsemilattice which intersects every ∘-class∘ exactly once. Given ( ) with content 𝑥𝑥1 𝑥𝑥𝑛𝑛 𝛽𝛽 𝑛𝑛 ≥ 𝑥𝑥1 𝑥𝑥𝑛𝑛 ∈ 𝑋𝑋 ( ) = { , … } we let ( ) be the unique element ( ) of this semilattice∘ + which is 𝒟𝒟 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 -related to in ( ) / ∘ : the mapping ( ∘ ) yields∘ an idempotent 𝑐𝑐 𝑤𝑤 𝑥𝑥1 𝑥𝑥𝑛𝑛 𝑤𝑤𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝛽𝛽 endomorphism which induces∘ +the -relation. The unary band thus∘ obtained will be denoted by . 𝒟𝒟 𝑤𝑤𝑤𝑤 𝑋𝑋 ∪ 𝑋𝑋 𝛽𝛽 𝑤𝑤𝑤𝑤 → 𝑤𝑤𝑤𝑤 For the sake of simplicity𝒟𝒟 we drop the notation and we use “=” to denote equality𝐹𝐹 in . Thus for , ( ) we write = (in ) instead of . We shall reserve “ ” for the equality in ( ∘ )+ . We shall denote the 𝛽𝛽semilattice transversal of consisting𝐹𝐹 of the𝑣𝑣 𝑤𝑤elements∈ 𝑋𝑋 ∪ 𝑋𝑋…∘ ∗ , 1, 𝑣𝑣 , …𝑤𝑤 𝐹𝐹 , by . Clearly𝑣𝑣𝑣𝑣𝑣𝑣 is a model of ≡ ( ): if : , then𝑋𝑋 for∪ ∘𝑋𝑋 every∘ and every mapping ∘ : , there exists𝐹𝐹 a 𝑥𝑥1 𝑥𝑥𝑛𝑛 𝑛𝑛 ≥ 𝑥𝑥1 𝑥𝑥𝑛𝑛 ∈ 𝑋𝑋 𝐹𝐹 𝐹𝐹 (unique)∘ homomorphism of unary bands ∘ : such that = . If is finite then so is 𝐁𝐁 𝐹𝐹 𝑋𝑋 , and𝜄𝜄 𝑋𝑋therefore→ 𝐹𝐹 is also finite,𝑆𝑆 ∈ since𝐁𝐁 finitely generated bands𝜑𝜑 𝑋𝑋 →are𝑆𝑆 finite. In other 𝜑𝜑 𝜄𝜄𝜑𝜑 words,∘ the variety is locally finite. From𝜑𝜑 this𝐹𝐹 → it follows𝑆𝑆 that there exists𝑋𝑋 an algorithm 𝑋𝑋 ∪ 𝑋𝑋 𝐹𝐹 + which decides whether∘ for given , , ( ) we have that = in F. We shall give an algorithm which𝐁𝐁 is transparent enough to be∘ useful. The algorithm which we set out to describe is similar to the algorithm𝑣𝑣 𝑣𝑣 given𝑤𝑤 ∈ in𝑋𝑋2∪ for𝑋𝑋 free bands. 𝑣𝑣 𝑤𝑤

We shall need some invariants. In our context, a property of a word which belongs to ( ) is called an invariant if whenever = in and satisfies this property, then so does∘ + . To “have the same content” is such an invariant: recall that for , , 𝑋𝑋 ∪ 𝑋𝑋+ 𝑣𝑣 𝑤𝑤 𝐹𝐹 𝑣𝑣 ( ) , ( ) = ( ) if and only if in , or if and only if in . ∘ 𝑤𝑤 𝑣𝑣 𝑣𝑣 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋For any𝑐𝑐 𝑣𝑣 (𝑐𝑐 𝑤𝑤 ) we define𝑣𝑣 𝑣𝑣𝑣𝑣 𝐹𝐹 to be the word obtained𝑣𝑣𝑣𝑣𝑣𝑣 from𝐹𝐹 by deleting first every occurrence of an ∘ ∗ in when preceded∗ somewhere in by or , and then 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑤𝑤𝐿𝐿 ∈ 𝑋𝑋 𝑤𝑤 deleting every occurrence of any . Thus for instance, for , , , ∘ 𝑥𝑥 ∈ 𝑋𝑋∘ 𝑤𝑤 ∘ 𝑤𝑤 𝑥𝑥 𝑥𝑥 𝑦𝑦( ∈ 𝑋𝑋 ) 1𝑥𝑥, 𝑦𝑦 𝑧𝑧 ∈ 𝑋𝑋

( ∘ ∘ ∘ ∘ ) . 𝐿𝐿 𝑥𝑥∘ 𝑦𝑦 𝑥𝑥∘𝑥𝑥∘𝑧𝑧 𝑦𝑦𝑦𝑦∘ ≡ The letters of such words 𝑥𝑥 𝑦𝑦𝑦𝑦 (if any)𝑥𝑥 𝑦𝑦 belong𝑧𝑧𝑧𝑧𝑧𝑧 to 𝐿𝐿 and≡ are𝑦𝑦𝑦𝑦 necessarily distinct. If 1, then the last letter of will be denoted by ( ). We can then uniquely write 𝑤𝑤𝐿𝐿 𝑋𝑋 𝑤𝑤𝐿𝐿 ≢ ( ) ( ) for some prefix ( ) of where ( ) ( ( )) and some suffix . Here ( ) or 𝑤𝑤𝐿𝐿 𝑙𝑙 𝑤𝑤 𝑤𝑤 ≡ may well be empty. Thus, for instance, 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑢𝑢 𝑙𝑙 𝑤𝑤 𝑤𝑤 𝑙𝑙 𝑤𝑤 ∉ 𝑐𝑐 𝑙𝑙 𝑤𝑤 𝑢𝑢 𝑙𝑙 𝑤𝑤 𝑢𝑢

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

( ) =

( ∘ ∘ ∘ ∘) = . 𝑙𝑙 𝑥𝑥∘𝑦𝑦𝑦𝑦𝑥𝑥∘𝑦𝑦∘𝑧𝑧𝑧𝑧𝑧𝑧∘ 𝑧𝑧 ∘ ∘ ∘ If = 1, it will be𝑙𝑙 convenient𝑥𝑥 𝑦𝑦𝑦𝑦𝑥𝑥 𝑦𝑦to put𝑧𝑧𝑧𝑧 𝑧𝑧( ) (𝑥𝑥)𝑦𝑦𝑦𝑦1𝑥𝑥. 𝑦𝑦 𝐿𝐿 In𝑤𝑤 a left-right dual way we define, for𝑙𝑙 𝑤𝑤 ≡( 𝑙𝑙 𝑤𝑤 ≡) , the word and, whenever 1, the variable ( ) and the suffix ( )∘ of∗ . Then if 1∗, 𝑅𝑅 ( ) ( ) in ( ) for some prefix .𝑤𝑤 If ∈ 𝑋𝑋 ∪1 𝑋𝑋we put ( ) 𝑤𝑤 ( ∈)𝑋𝑋 1. 𝑅𝑅 𝑅𝑅 𝑤𝑤 ≢ ∘ ∗ 𝑟𝑟 𝑤𝑤 𝑟𝑟 𝑤𝑤 𝑤𝑤 𝑤𝑤 ≢ 𝑤𝑤 ≡ 𝑅𝑅 𝑣𝑣Lemma𝑟𝑟 𝑤𝑤 𝑟𝑟 2.1𝑤𝑤 𝑋𝑋 ∪ 𝑋𝑋 𝑣𝑣 𝑤𝑤 ≡ 𝑟𝑟 𝑤𝑤 ≡ 𝑟𝑟 𝑤𝑤 ≡

For ( ) with ( ) = { , … , }, ∘ + 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑐𝑐 𝑤𝑤 =𝑥𝑥1 𝑥𝑥𝑛𝑛 in F if 1,

∘ ∘ = (∘ ) (∘ ) in F otherwise. 𝑤𝑤 ℛ𝑤𝑤𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝑤𝑤𝐿𝐿 ≡ 𝑎𝑎𝑎𝑎𝑎𝑎 ∘ ∘ ∘ ∘ Proof 𝑤𝑤 ℛ𝑤𝑤𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛

If ( ) with ( ) = { , … , }, then or where∘ + and for+ some . … , … , 1 , 𝑛𝑛… ( ) 1 ∘ 𝑤𝑤 ∈∘ 𝑋𝑋 ∪ 𝑋𝑋 𝑐𝑐 𝑤𝑤 ∗𝑥𝑥 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 𝑤𝑤 ≡ 0 1 1 𝑘𝑘 𝑘𝑘 0 𝑘𝑘 1 𝑘𝑘 𝑤𝑤 𝑦𝑦 𝑤𝑤First𝑦𝑦 we𝑤𝑤 assume that𝑤𝑤 𝑤𝑤 1∈. Then𝑋𝑋 can𝑦𝑦 be𝑦𝑦 written∈ 𝑐𝑐 𝑤𝑤as = 𝑘𝑘 ≥ , 1, where ( ) = { , … , } such that, for every 1 , we have that∘ ( )∘ 𝐿𝐿 1 1 𝑘𝑘 𝑘𝑘 { , … , }. We prove by induction𝑤𝑤 ≡ that for𝑤𝑤 all 1 we have𝑤𝑤 that𝑦𝑦 𝑤𝑤 =⋯ 𝑦𝑦 𝑤𝑤 𝑘𝑘 ≥ in 𝑐𝑐 𝑤𝑤 𝑦𝑦1 𝑦𝑦𝑘𝑘 ≤ 𝑖𝑖 ≤ 𝑘𝑘 𝑐𝑐 𝑤𝑤𝑖𝑖 ⊆ . This is obviously true for = 1. Suppose that for < we have that ∘ ∘ 𝑦𝑦1 𝑦𝑦𝑖𝑖 ≤ 𝑖𝑖 ≤ 𝑘𝑘 𝑤𝑤 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑤𝑤 𝐹𝐹 = 𝑖𝑖= 𝑖𝑖 𝑘𝑘 . ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ Since 𝑤𝑤 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑤𝑤 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑦𝑦1𝑤𝑤1𝑦𝑦2𝑤𝑤2 ⋯ 𝑦𝑦𝑖𝑖 𝑤𝑤𝑖𝑖𝑦𝑦𝑖𝑖+1 ⋯ 𝑦𝑦𝑘𝑘𝑤𝑤𝑘𝑘 ( ) = , … , = ( ), ∘ ∘ ∘ ∘ ∘ ∘ it follows that 𝑐𝑐 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑦𝑦1𝑤𝑤1 ⋯ 𝑦𝑦𝑖𝑖 𝑤𝑤𝑖𝑖 𝑦𝑦1 𝑦𝑦𝑖𝑖 𝑐𝑐 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖

∘ ∘ ∘ ∘ ∘ ∘ in . Therefore 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 ℛ𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑦𝑦1𝑤𝑤1 ⋯ 𝑦𝑦𝑖𝑖 𝑤𝑤𝑖𝑖 𝐹𝐹 Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 6

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

∘ ∘ ∘ ∘ ∘ = ∘ ∘ ∘ ∘ ∘ ∘ , 1 𝑖𝑖 1 1 𝑖𝑖 𝑖𝑖 𝑖𝑖+1 𝑖𝑖+1 1 𝑖𝑖 1 1 𝑖𝑖 𝑖𝑖 𝑖𝑖+1 𝑦𝑦 ⋯ 𝑦𝑦 𝑦𝑦 𝑤𝑤 ⋯ 𝑦𝑦 𝑤𝑤 𝑦𝑦 𝑦𝑦∘ 𝑦𝑦 ∘⋯∘𝑦𝑦 𝑦𝑦∘ 𝑤𝑤 ⋯ 𝑦𝑦∘ 𝑤𝑤 𝑦𝑦∘ since the elements of which are -related to an𝑦𝑦 element1 ⋯ 𝑦𝑦 𝑖𝑖of𝑦𝑦 the𝑖𝑖+ 1semilattice𝑦𝑦1𝑤𝑤1 ⋯ transversal𝑦𝑦𝑖𝑖 𝑤𝑤𝑖𝑖𝑦𝑦𝑖𝑖+ 1 of form a right regular band by Result 1.1. It follows that = . Using induction∘ and 𝐹𝐹 ℛ 𝐹𝐹 𝐹𝐹 the fact that = we find that = ∘ . Since∘ ∘ in , we thus 𝑤𝑤 𝑦𝑦1 ⋯ 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖+1𝑤𝑤 have that ∘ ∘ =∘ ∘ = ∘ ∘. ∘ ∘ 1 𝑛𝑛 1 𝑘𝑘 1 𝑛𝑛 1 𝑛𝑛 𝑥𝑥 ⋯∘𝑥𝑥 ∘ 𝑦𝑦 ⋯∘𝑦𝑦 ∘ ∘ 𝑤𝑤∘ 𝑥𝑥∘ ⋯ 𝑥𝑥∘ 𝑤𝑤 𝑤𝑤𝑤𝑤𝑥𝑥 ⋯ 𝑥𝑥 𝐹𝐹 We𝑤𝑤 nextℛ𝑤𝑤 𝑥𝑥consider1 ⋯ 𝑥𝑥𝑛𝑛 the 𝑥𝑥case1 ⋯ where𝑥𝑥𝑛𝑛𝑤𝑤𝑥𝑥 1 ⋯ 𝑥𝑥1𝑛𝑛. Then𝑥𝑥1 there⋯ 𝑥𝑥𝑛𝑛 exists ( ) such that ( ) ( ) for some ( ), ( ) . If ( ) = { , , }, we may as well assume 𝑤𝑤𝐿𝐿 ≢ 𝑙𝑙 𝑤𝑤 ∈ 𝑋𝑋 𝑤𝑤 ≡ that ( ( ) ( )) = { , , } for some∘ ∗ . Then 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑢𝑢 𝑙𝑙 𝑤𝑤 𝑢𝑢 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑐𝑐 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝑐𝑐 𝑙𝑙 =𝑤𝑤 𝑙𝑙( 𝑤𝑤) ( )(𝑥𝑥1( ⋯) (𝑥𝑥𝑗𝑗 )) ( ) 𝑗𝑗( ≤)𝑛𝑛 (since ( ) ( ) ( ( ) ( )) in ) = ( ) ( ) ( )∘ ( ) ∘ 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑢𝑢 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝒟𝒟 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝐹𝐹 = ( ) ( ) ∘ ∘ 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑗𝑗 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑢𝑢 where 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤( 𝑣𝑣) ( ) is such that = 1 and ( ) = { , … , } = ( ). By the ∘ ∘ first part of the proof, = … and thus = ( ) ( ) . Since and 𝑣𝑣 ≡ 𝑥𝑥1 ⋯ 𝑥𝑥𝑗𝑗 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑢𝑢 𝑣𝑣𝐿𝐿 𝑐𝑐 𝑣𝑣 𝑥𝑥1 𝑥𝑥𝑛𝑛 𝑐𝑐 𝑤𝑤 ( ) ( ) have the∘ same∘ content, they must then be -related∘ ∘ in . Further, 𝑣𝑣 𝑥𝑥1 𝑥𝑥𝑛𝑛𝑣𝑣 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛𝑣𝑣 𝑤𝑤 ∘ ∘ = and so = ( ) ( ) . 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 ℛ 𝐹𝐹 ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑥𝑥1 ⋯ 𝑥𝑥From𝑣𝑣𝑥𝑥1 ⋯Lemma𝑥𝑥 2.1𝑥𝑥1 and⋯ 𝑥𝑥 its dual we𝑤𝑤𝑥𝑥 have1 ⋯ 𝑥𝑥 the following.𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥 □

Corollary 2.2

Let ( ) , ( ) = { , … , }. ∘ + 1 𝑛𝑛 If 𝑤𝑤 ∈1𝑋𝑋 ∪ 𝑋𝑋, then𝑐𝑐 𝑤𝑤 𝑥𝑥 𝑥𝑥

𝐿𝐿 𝑅𝑅 𝑤𝑤 ≢ ≢(𝑤𝑤) ( ) ( ) ( ) ∘ ∘ ∘ ∘ ∘ ∘ and = 𝑤𝑤( ) ℛ( 𝑙𝑙) 𝑤𝑤 𝑙𝑙 𝑤𝑤 (𝑥𝑥1)⋯( 𝑥𝑥)𝑛𝑛. ℒ𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛ℛ𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛𝑟𝑟 𝑤𝑤 𝑟𝑟 𝑤𝑤 ℒ𝑤𝑤 ∘ ∘ If 𝑤𝑤 1𝑙𝑙 𝑤𝑤 𝑙𝑙 ,𝑤𝑤 then𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛𝑟𝑟 𝑤𝑤 𝑟𝑟 𝑤𝑤 𝑤𝑤=𝐿𝐿 ≢( )≡( 𝑤𝑤)𝑅𝑅 .. ∘ ∘ ∘ ∘ 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑤𝑤 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛ℒ𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 7

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

If 1 , then

𝑤𝑤=𝐿𝐿 ≡ ≢ 𝑤𝑤𝑅𝑅( ) ( ) . ∘ ∘ ∘ ∘ 𝑤𝑤If 𝑥𝑥1 1⋯ 𝑥𝑥𝑛𝑛𝑟𝑟 𝑤𝑤, then𝑟𝑟 𝑤𝑤 =ℛ𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 . ∘ ∘ 𝐿𝐿 𝑅𝑅 1 𝑛𝑛 That𝑤𝑤 the≡ ,≡¯, 𝑤𝑤 and yield𝑤𝑤 invariants𝑥𝑥 ⋯ 𝑥𝑥 follows from the following sequence of results.

Lemma 2.3𝑙𝑙 𝑙𝑙 𝑟𝑟 𝑟𝑟

Let , ( ) be such that = in . Then 1 if and only if 1. If this is the case, then ( ∘) + ( ) and either ( ) 1 ( ) or ( ) = ( ) in . 𝑣𝑣 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑣𝑣 𝑤𝑤 𝐹𝐹 𝑣𝑣𝐿𝐿 ≢ 𝑤𝑤𝐿𝐿 ≢ Proof 𝑙𝑙 𝑣𝑣 ≡ 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑣𝑣 ≡ ≡ 𝑙𝑙 𝑤𝑤 𝑙𝑙 𝑣𝑣 𝑙𝑙 𝑤𝑤 𝐹𝐹

If = in then can be transformed into w by using a finite sequence of transformations of the form where , ( ) and where ( , ) or ( , ) is one𝑣𝑣 of𝑤𝑤 the pairs𝐹𝐹 listed𝑣𝑣 in (2). It therefore suffices to prove∘ ∗ the statement of the theorem for the special case 𝑝𝑝𝑝𝑝𝑝𝑝where→ 𝑝𝑝𝑝𝑝𝑝𝑝 and𝑝𝑝 𝑞𝑞 ∈ 𝑋𝑋 ∪ where𝑋𝑋 ( , ) is any 𝑠𝑠of𝑡𝑡 the pairs listed𝑡𝑡 𝑠𝑠 in (2). The proof is not much harder than the corresponding verification for free bands as in the proof of Lemma 4.5.1𝑣𝑣 of≡.3𝑝𝑝𝑝𝑝𝑝𝑝 𝑤𝑤 ≡ 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠 𝑡𝑡

Using an inductive argument we thus□ have the following.

Corollary 2.4

If , ( ) are such that = in , then . ∘ + 𝐿𝐿 𝐿𝐿 It 𝑣𝑣will𝑤𝑤 be∈ convenient𝑋𝑋 ∪ 𝑋𝑋 to denote by 𝑣𝑣 the𝑤𝑤 unary𝐹𝐹 band𝑣𝑣 with≡ 𝑤𝑤 an identity 1 adjoined. The unary operation of can be extended to1 by putting 1 = 1. From Corollaries 2.2 and 2.4, Lemma 2.3 and their duals, we conclude𝐹𝐹 1 the following.𝐹𝐹∘ 𝐹𝐹 𝐹𝐹 Theorem 2.5

For , ( ) we have: if = in , then ∘ + 𝑣𝑣 𝑤𝑤 ∈ 𝑋𝑋 (∪ 𝑋𝑋) = ( ), 𝑣𝑣 𝑤𝑤 𝐹𝐹 , ,

and 𝑐𝑐 𝑣𝑣 𝑐𝑐 𝑤𝑤 𝑣𝑣𝐿𝐿 ≡ 𝑤𝑤𝐿𝐿 𝑣𝑣𝑅𝑅 ≡ 𝑤𝑤𝑅𝑅 Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 8

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

( ) = ( ) in if 1,

( ) = ( ) 1 1. in if 𝐿𝐿 𝐿𝐿 𝑙𝑙 𝑣𝑣 𝑙𝑙 𝑤𝑤 𝐹𝐹1 𝑣𝑣 ≡ 𝑤𝑤 ≢ Conversely, if ( ) = 𝑟𝑟( 𝑣𝑣), then 𝑟𝑟= 𝑤𝑤 in if𝐹𝐹 either𝑣𝑣 one𝑅𝑅 ≡of the𝑤𝑤𝑅𝑅 following≢ occur:

1. (i) 𝑐𝑐 𝑣𝑣 𝑐𝑐1𝑤𝑤, 𝑣𝑣 𝑤𝑤1, (𝐹𝐹) = ( ) in and ( ) = ( ) in , 1 1 𝐿𝐿 𝐿𝐿 𝑅𝑅 𝑅𝑅 (ii)𝑣𝑣 ≡ 𝑤𝑤 ≢ 1𝑣𝑣, ≡ 𝑤𝑤 ≢ 1𝑙𝑙, and𝑣𝑣 ( 𝑙𝑙)𝑤𝑤= (𝐹𝐹) in 𝑟𝑟, 𝑣𝑣 𝑟𝑟 𝑤𝑤 𝐹𝐹 1 (iii)𝑣𝑣 𝐿𝐿 ≡ 𝑤𝑤𝐿𝐿 ≡ 1,𝑣𝑣 𝑅𝑅 ≡ 𝑤𝑤𝑅𝑅 ≢ 1, and𝑟𝑟 (𝑣𝑣) = 𝑟𝑟(𝑤𝑤) in 𝐹𝐹 , 1 (iv) 𝑣𝑣𝐿𝐿 ≡ 𝑤𝑤𝐿𝐿 ≢1 𝑣𝑣𝑅𝑅 ≡ 𝑤𝑤𝑅𝑅 ≡. 𝑙𝑙 𝑣𝑣 𝑙𝑙 𝑤𝑤 𝐹𝐹 𝑣𝑣𝐿𝐿 ≡ 𝑤𝑤𝐿𝐿 ≡ ≡ 𝑣𝑣𝑅𝑅 ≡ 𝑤𝑤𝑅𝑅

□ We remark here that the task for testing whether = in is, by Theorem 2.5, reduced to a similar task for words of smaller content. Proceeding inductively, Theorem 2.5 thus allows us to verify whether = in in a finite number𝑣𝑣 𝑤𝑤 of steps.𝐹𝐹 We shall be more explicit in the following. 𝑣𝑣 𝑤𝑤 𝐹𝐹 Let { , } be the free monoid on the set containing the symbols and , and again let 1 stand for the identity∗ element of { , } . For { , } and ( ) , we define ( ) inductively by:𝑙𝑙 𝑟𝑟 ∗ ∗ 𝑙𝑙 ∘ ∗𝑟𝑟 𝑙𝑙 𝑟𝑟 𝑘𝑘 ∈ 𝑙𝑙 𝑟𝑟 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑘𝑘 𝑤𝑤 1( ) =

and𝑤𝑤 for 𝑤𝑤= with , … , { , }, 1, ( ) =𝑘𝑘 (𝑘𝑘1 ⋯…𝑘𝑘𝑛𝑛 ( ))𝑘𝑘. 1 𝑘𝑘𝑛𝑛 ∈ 𝑙𝑙 𝑟𝑟 𝑛𝑛 ≥ 𝑘𝑘 𝑤𝑤 The𝑘𝑘 1following𝑘𝑘2 𝑘𝑘 𝑛𝑛follows𝑤𝑤 from Theorem 2.5 using induction. For { , } , we let | | be the length of , and, for a set , we let | | be the cardinality of . ∗ 𝑘𝑘 ∈ 𝑙𝑙 𝑟𝑟 𝑘𝑘 Theorem𝑘𝑘 2.6 𝐴𝐴 𝐴𝐴 𝐴𝐴

For , ( ) we have that = in if and only if ( ) = ( ) and for any { , } with | | |∘ (+ )| = | ( )|, 𝑣𝑣∗𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑣𝑣 𝑤𝑤 𝐹𝐹 𝑐𝑐 𝑣𝑣 𝑐𝑐 𝑤𝑤 𝑘𝑘 ∈ 𝑙𝑙 𝑟𝑟 𝑘𝑘 ≤ 𝑐𝑐 𝑣𝑣 𝑐𝑐 𝑤𝑤

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

( ( )) = ( ( )),

( ) ( ) and ( ) ( ) . 𝑐𝑐 𝑘𝑘 𝑣𝑣 𝑐𝑐 𝑘𝑘 𝑤𝑤 𝑘𝑘 𝑣𝑣 𝐿𝐿 ≡ 𝑘𝑘 𝑤𝑤 𝐿𝐿 𝑘𝑘 𝑣𝑣 𝑅𝑅 ≡ 𝑘𝑘 𝑤𝑤 𝑅𝑅 □ From Theorem 2.6 and the solution of the word problem for the free band ( ) on the set (see [2] or Section 4.5 in [3]) it follows that for , , and represent the 𝐁𝐁 same element of the free band ( ) if and only if = in . We+ can state this as𝐹𝐹 in𝑋𝑋 the following𝑋𝑋 corollary. 𝑣𝑣 𝑤𝑤 ∈ 𝑋𝑋 𝑣𝑣 𝑤𝑤 𝐹𝐹𝐁𝐁 𝑋𝑋 𝑣𝑣 𝑤𝑤 𝐹𝐹 Corollary 2.7

Let be a nonempty set. Then the mapping , , can be extended to an embedding of the free band ( ) as a subband of = ∘ ( ). 𝑋𝑋 𝑋𝑋 ↦ 𝑋𝑋 ∪ 𝑋𝑋 ∘𝑥𝑥 ↦ 𝑥𝑥 𝐁𝐁 𝐁𝐁 In other words, any free band𝐹𝐹 can𝑋𝑋 be isomorphically𝐹𝐹 embedded𝐹𝐹 𝑋𝑋 into a band which has a semilattice transversal. Therefore, the free objects in the quasivariety consisting of the bands which can be embedded into some band which has a semilattice transversal, are the familiar free bands.

Example 1

The -classes of = ( ) are much larger than the corresponding -classes of the free band ( ). Thus, if and∘ are distinct elements of , then the -class of in 𝐁𝐁 consists𝒟𝒟 of the 11𝐹𝐹 elements𝐹𝐹 𝑋𝑋 , , , , , 𝒟𝒟 ∘ ,∘ , 𝐁𝐁 𝐹𝐹 , 𝑋𝑋 , 𝑥𝑥 𝑦𝑦, ∘and∘ ∘ ∘ ∘ . Therefore∘ ∘ ∘𝑋𝑋 the∘ ∘-classℒ∘ of∘ ∘ in𝑥𝑥 ∘ 𝑦𝑦 ∘contains∘ 𝐹𝐹 ∘ ∘ exactly∘ ∘ 121 elements.∘ ∘ ∘ The∘ ∘𝑥𝑥-class𝑦𝑦 𝑥𝑥 𝑥𝑥 ∘ 𝑦𝑦in the∘𝑦𝑦 ∘𝑥𝑥 free𝑥𝑥 𝑦𝑦 band𝑦𝑦 𝑦𝑦𝑦𝑦 𝑥𝑥( 𝑦𝑦) contains𝑦𝑦𝑥𝑥 𝑦𝑦 𝑥𝑥only𝑦𝑦𝑥𝑥 the𝑦𝑦 4𝑥𝑥 elements𝑥𝑥𝑥𝑥𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥,𝑥𝑥 𝑦𝑦 ,𝑥𝑥𝑥𝑥 , 𝑦𝑦and𝑥𝑥 𝑦𝑦 𝑦𝑦𝑦𝑦. 𝑥𝑥 𝑦𝑦 𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 𝑦𝑦 𝒟𝒟 𝑥𝑥𝑥𝑥 𝐹𝐹 𝒟𝒟 𝑥𝑥𝑥𝑥 𝐹𝐹𝐁𝐁 𝑋𝑋 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥If 𝑦𝑦𝑦𝑦= { ,𝑦𝑦𝑦𝑦𝑦𝑦} then the 11 elements listed above, together with , , , and form the left regular band which is a transversal of the -classes of ∘( ). ∘The∘ 6-element∘ 𝑋𝑋 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑦𝑦 𝑦𝑦𝑦𝑦 free band ( ) is a normal band, whereas the 129-element free split ∘band ( ) is not: ℛ 𝐹𝐹𝐁𝐁 𝑋𝑋 ( ) is not a semilattice, but a 9-element (square) rectangular band with∘ an 𝐹𝐹𝐁𝐁 𝑋𝑋 𝐹𝐹𝐁𝐁 𝑋𝑋 identity∘ ∘ adjoined.∘ If is the smallest fully invariant congruence on ( ) which identifies 𝑥𝑥 𝐹𝐹𝐁𝐁 𝑋𝑋 𝑥𝑥 these 9 elements, then separates the elements of ( ): the free normal∘ band ( ) is 𝐁𝐁 embeddable into the𝜌𝜌 split band ( )/ , which is a normal band.𝐹𝐹 𝑋𝑋 𝐁𝐁 𝐁𝐁 𝜌𝜌 ∘ 𝐹𝐹 𝑋𝑋 𝐹𝐹 𝑋𝑋 𝐁𝐁 Semigroup forum, Vol. 90, No. 3 (June 2015): pg. 7𝐹𝐹53-762𝑋𝑋. DOI. This𝜌𝜌 article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 10

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

3 Normal forms

We continue to use the notation introduced in the first few paragraphs of the previous section. Thus is the congruence relation on ( ) generated by the pairs in (2), and as we have seen ( ) / yields the free split band∘ + ( ) on the set . We 𝛽𝛽 𝑋𝑋 ∪ 𝑋𝑋 would like to find a subset of ( ∘ + ) which is a cross section of the∘ -classes. The 𝐁𝐁 elements of this subset are𝑋𝑋 then∪ 𝑋𝑋 called∘𝛽𝛽 of+ normal form, and every 𝐹𝐹 𝑋𝑋( ) is𝑋𝑋 related to a unique element of 𝑋𝑋normal∪ 𝑋𝑋 form, called the normal form of w𝛽𝛽. We would∘ + like to do all this in such a way that, given any ( ) , there exists𝑤𝑤 ∈ an 𝑋𝑋algorithm∪ 𝑋𝑋 which𝛽𝛽 produces the normal form of . After this is accomplished,∘ + we have a concrete model of ( ) at hand. 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 ∘ 𝑤𝑤 𝐁𝐁 𝐹𝐹 𝑋𝑋 In order to be able to find a normal form for the elements of ( ), we shall assume that there exists a total order on the set of generators. For any∘ finite nonempty 𝐁𝐁 set of , say = { , … , } with < … < for the total order,𝐹𝐹 𝑋𝑋 we let be the uniquely defined word of ( ) . In particular,𝑋𝑋 if ( ) , then (∘ ) 1 𝑛𝑛 1 2 𝑛𝑛 ( 𝐶𝐶) is𝑋𝑋 uniquely𝐶𝐶 defined𝑥𝑥 ∘ and𝑥𝑥 ∘ ( ( 𝑥𝑥∘) +) =𝑥𝑥 ( ). 𝑥𝑥 ∘ + 𝐶𝐶 ∘ 1 𝑛𝑛 ∘ + 𝑥𝑥 ⋯ 𝑥𝑥 𝑋𝑋 ∘ 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 𝑐𝑐 𝑤𝑤 ∈ 𝑋𝑋 From the results of the previous𝑐𝑐 𝑐𝑐 𝑤𝑤 section,𝑐𝑐 𝑤𝑤 the following holds.

Lemma 3.1

For any ( ) , = ( ) ( ) ( ) ( ) ( ). ∘ + ∘ It follows𝑤𝑤 that∈ we𝑋𝑋 can,∪ 𝑋𝑋 in a unique𝑤𝑤 𝑙𝑙way,𝑤𝑤 parse𝑙𝑙 𝑤𝑤 𝑐𝑐every𝑤𝑤 𝑟𝑟 𝑤𝑤(𝑟𝑟 𝑤𝑤 ) following the rule given by Lemma 3.1. We may visualize our parsing of by considering a ∘tree+ with a root labeled , 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 and where every node labeled ( ) is expanded as in Fig. 1. ∘ + 𝑤𝑤 𝑤𝑤 𝑣𝑣 ∈ 𝑋𝑋 ∪ 𝑋𝑋

Fig. 1 Expansion of node

Semigroup forum, Vol. 90, No. 3 (Jun𝑣𝑣e 2015): pg. 753-762. DOI. This article is © Springer and permission has been granted for this version to appear in e-Publications@Marquette. Springer does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer. 11

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

We shall, however, not consider such a further expansion if , 1 or ( ) . In the latter case we rewrite as ( ) . When parsing any ( ) we thus obtain∘ + a labeled tree whose leaves are labeled∘ with either 1, or a letter𝑣𝑣 ∈ of𝑋𝑋 𝑣𝑣, or≡ a∘ word+ 𝑣𝑣 ∈of (𝑋𝑋 ) . 𝑣𝑣 𝑐𝑐 𝑣𝑣 𝑤𝑤 ∈ 𝑋𝑋 ∪ 𝑋𝑋 ∘ + 𝑋𝑋 𝑋𝑋 By way of example, we give the parsing tree for the word with < < < < < in X: see Fig. 2. The corresponding 𝑤𝑤 ≡ normal∘ ∘form∘ for ∘ may∘ then be calculated, using Lemma 3.1, as 𝑦𝑦𝑥𝑥 𝑦𝑦𝑞𝑞 𝑥𝑥𝑠𝑠 𝑦𝑦𝑦𝑦𝑦𝑦𝑡𝑡 𝑧𝑧 𝑠𝑠 𝑞𝑞 𝑠𝑠 𝑡𝑡 𝑥𝑥 𝑦𝑦 𝑧𝑧 = [ 𝑤𝑤 ] ( ) [ ] = 1 ∘ (∘ ∘ ) ∘ [∘ ∘ ]∘ ∘ ∘( ∘ )∘ ∘ [ ] 𝑤𝑤 𝑦𝑦𝑥𝑥( 𝑦𝑦𝑞𝑞 𝑥𝑥∘𝑠𝑠∘𝑦𝑦)∘ ⋅ ∘𝑧𝑧 ⋅ [𝑞𝑞 𝑠𝑠 𝑡𝑡∘] 𝑥𝑥 𝑦𝑦 𝑧𝑧 ⋅∘ 𝑥𝑥∘⋅ ∘𝑠𝑠 ∘𝑦𝑦𝑦𝑦𝑦𝑦∘ ∘𝑡𝑡 𝑧𝑧 𝑠𝑠 ∘ = 1 ⋅ 𝑦𝑦 ⋅∘ (∘𝑞𝑞 ∘𝑠𝑠 𝑥𝑥∘ 𝑦𝑦 ) ⋅ 𝑥𝑥∘⋅ ∘𝑠𝑠 𝑦𝑦 ⋅ 𝑧𝑧( ⋅ 𝑞𝑞)𝑠𝑠 𝑡𝑡 𝑥𝑥1𝑦𝑦 𝑧𝑧 (⋅ 𝑥𝑥 ⋅ 𝑠𝑠 𝑦𝑦 ⋅ ) 𝑧𝑧 ⋅ 𝑠𝑠 𝑡𝑡(𝑦𝑦∘ 𝑧𝑧∘ )∘ ⋅ 𝑦𝑦∘ ⋅ 1𝑡𝑡 𝑧𝑧 ∘𝑠𝑠( ∘ ∘) ( ) ∘ (∘ ∘ ∘ ∘) ∘ 1. ∘⋅ 𝑦𝑦 ⋅ 𝑞𝑞 ∘𝑠𝑠 ∘𝑥𝑥 𝑦𝑦 ⋅ 𝑥𝑥 ⋅ 𝑠𝑠 ⋅ 𝑦𝑦∘ ⋅∘ 𝑠𝑠∘ 𝑦𝑦∘ ⋅ 𝑦𝑦 ⋅ ∘⋅ 𝑧𝑧∘ ⋅ 𝑞𝑞 𝑠𝑠 𝑡𝑡∘𝑥𝑥∘ 𝑦𝑦∘ 𝑧𝑧 ⋅ 𝑥𝑥 ⋅ Here the𝑠𝑠 ⋅square𝑦𝑦 ⋅ 𝑠𝑠 brackets𝑦𝑦 ⋅ 𝑦𝑦 indicate⋅ ⋅ 𝑧𝑧 that⋅ 𝑠𝑠 further𝑡𝑡 𝑦𝑦 𝑧𝑧 expansion⋅ 𝑦𝑦 ⋅ 𝑡𝑡 is𝑧𝑧 required,⋅ 𝑠𝑠 ⋅ and𝑠𝑠 𝑡𝑡 dots𝑧𝑧 suggest⋅ 𝑠𝑠 ⋅ that some branching in the parsing tree is involved.

Fig. 2 Parsing of the word w

References

1Gerhard, J.A.: The lattice of equational classes of idempotent semigroups. J. Algebra 15, 195–224 (1970) 2Green, J.A., Rees, D.: On semigroups in which = . Proc. Camb. Philos. Soc. 48, 35–40 (1952) 3Howie, J.M.: Fundamentals of Semigroup Theory𝑟𝑟 . Clarendon Press, Oxford (1995) 4McAlister, D.B., Blyth, T.S.: Split orthodox semigroups.𝑥𝑥 𝑥𝑥 J. Algebra 51, 491–525 (1978) 5McLean, D.: Idempotent semigroups. Am. Math. Mon 61, 110–113 (1954) 6Petrich, M.: Lectures in Semigroups. Akademie-Verlag, Berlin (1977)

NOT THE PUBLISHED VERSION; this is the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation at the bottom of the page.

7Saito, T.: A note on regular semigroups with inverse transversals. Semigroup Forum 33, 149–152 (1986) 8Tang, X.: LRT-biordered sets. Semigroup Forum 73, 377–394 (2006) 9Yoshida, R.: Right regular bands with semilattice transversals; In: Proceedings of the 8th Symposium on Semigroups, Shimane University, Matsue, Pp. 26–31 (1984)