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Free Split Bands Francis Pastijn Marquette University, Francis.Pastijn@Marquette.Edu 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 bands S for which the least semilattice congruence splits, that is, there exists a subsemilattice of which intersects each -class in exactly one element. Such a subsemilattice ∘ of will then be called a semilattice transversal of . ∘ 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∘ ∘ endomorp hism of which induces the -relation on , and in∘ particular, ∈ is a ∈ ↦ ↦ retract of . The unary band (( , , )) thus obtained obviously satisfies, apart from the associative∘ law, the identities ∘ ⋅ , , , ( ) 2 ∘ , (∘ )∘ ∘ . ≈ ≈ ≈ (1) ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ≈ ≈ ≈ 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 = ≈ { | } is a semilattice transversal of the band∘ ( , ). For this reason we call the unary∘ bands which∘ satisfy the identities (1) split bands. ⋅ ∈ ⋅ 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 and , respectively.∘ As the abstract indicates, the∘ purpose of this paper is to solve the word problem∘ for ( ) and to show that FB(X) can be isomorphically embedded into the multiplicative reduct∘ of ( ). The solution of the word problem for ( ) is well understood and the reader will∘ find all the necessary details in Sect 4.5 of3 where appropriate references to1 and 2 are given. Our solution to the word problem for ( ) is similar though slightly more complicated. While is, like , locally finite, for any finite∘ nonempty set the (finite) free object ( ) is much∘ larger than ( ). ∘ There is something more enigmatic about all this. It turns out that 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 [ - classes] of . This∘ follows from a very special applicationℒ of Propositionℛ 2.3 and Corollary 2.4 of8 and below we intend to give a short independent proof ℛof this fact ℒin the special circumstances 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 is∘ a transversal ⋅ of the -classes [ -classes] of ( , ). ∈ ∘ ∈ ⋅ PΛroof ℛ ℒ ⋅ 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∘ ∈ ℒ a sub∘semigroup of , or in particular, that for any , we have that∘ ( )( ) ( ) . In ℛ any case = = ( ) is -related to ( )( ) in and since∘ this ∘-class is a∘ ∈ ℒ rectangular∘ band∘ it follows∘ ∘ that (∘ )( ) ( ∘ )( ∘ ). Applying the identities (1) we find that ∘ ∘ ∘ ∘ ∘ ∘ ℒ ( )( ) = ( ) ∘ ∘ ∘ ∘ = ∘ ∘ ∘ ∘ = ∘ ∘ ∘ = ∘∘=∘( ) . ∘∘ ∘ Thus ( )( ) ( ) as required. ∘ ∘ ∘ The and ℒ mentioned 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 theorem of split bandsΛ in terms of the left and right regular bands and akin to, but simpler than the construction in II.1 of.6 For a free split band = ( ), it is easy to Λ characterize the elements of the left [right] regular split band (see∘ Corollary 2.2 and Theorem 2.5). By left-right duality, is anti-isomorphic to . Λ The variety of right [left] regular split bandsΛ is the subvariety of determined by the additional∘ identity∘ . Thus, with the notation of ∘ Result 1.1, and belong to and , respectively. As we shall see, if = ≈ ≈ ( ) is a free split band, then and∘ should∘ not be assumed to be free on in Λ and∘ , respectively. ∘ ∘ Λ It 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 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. 3 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, 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 introduces a variety of unary semigroups (whose members are all regular semigroups) which contains as a subvariety. ∘ 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. ∘ ∘ We let be a set disjoint of and , , a bijection. The elements of will be called letters∘ . The identity of the free monoid∘ ( ∘ ) is the empty word 1, thus ( ∘ ↦ ↦ ∪ ) = ( ) {1}. For any ( ) we define∘ ∗ the content ( ) of inductively∘ ∗ by ∘ + ∘ ∗ ∪ ∪ ∪ ∪ ∈ ∪ (1) = , ( ) = ( ) = { }, , ( ) = ∅( ∘ ) ( ), > 1, , …, ∈ . 1 −1 1 −1 ⋯ ⋯ ∪ ∘ We let be the congruence relation of the free semigroup1 (∈ ∪) generated by the pairs ∘ + ∪ ( , ), ( ) , ( , 2), ( , ), ∘ + , ( ∘ , ∘ ),∘ ∈∘ , ∪ . ∈ (2) ∘ ∘ ∘ ∘ ∈ 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 ∘ ∪ ( ) = ( ). (3) 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. 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 1 ≥ 1 ∈ ( ) = { , … } we let ( ) be the unique element ( ) of this semilattice∘ + which is ∈ ∪ -related to in ( )
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