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The word problem in regular band free products. (English) Zbl 0614.20041

At the time of review, the problem of finding a normal form for, or otherwise describing, the elements in the free product of two bands U and V, in the variety \({\mathcal B}\) of all bands, is unsolved. The author shows here, however, that at least for the free product within the variety \({\mathcal R}{\mathcal B}\) of ”regular” bands - those satisfying the identity \(xyxzx=xyzx\)- such a normal form exists. Any member of this free product has as normal form either \[ (i)\quad...v_{p-1}u_{t-1}v_ pu_ tv_{p+1}u_{p+1}v_{p+2}..., \] where \(u_{t-1},u_ t\) and \(u_ t,u_{t+1}..\). belong to strictly descending and strictly ascending chains of \({\mathcal D}\)-classes of U, respectively, and similarly for \(...v_{p-1}\), \(v_ p\) and \(v_ p,v_{p+1}..\). in V (except that possibly \(v_ p=v_{p+1})\), or (ii) a form obtained by interchanging the roles of U and V in (i). If U and V are semilattices this normal form is unique and in fact describes the free product of U and V in \({\mathcal B}\), by a result of the reviewer [Semigroup Forum 20, 335-341 (1980; Zbl 0448.20056)]. Otherwise uniqueness is ensured by the imposition of further restrictions.
Reviewer: P.R.Jones

MSC:

20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 0448.20056
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References:

[1] Howie, J.M.,An Introduction to Semigroup Theory, Academic Press, London, 1976. · Zbl 0355.20056
[2] Jones, P.R.,A band generated by two semilattices is regular, Semigroup Forum 20 (1980), 335–341. · Zbl 0448.20056 · doi:10.1007/BF02572693
[3] Olin, P.,Varietal free products of bands, Semigroup Forum 21 (1980), 83–87. · Zbl 0453.20049 · doi:10.1007/BF02572538
[4] Olin, P.,Normal forms for band free products, Semigroup Forum, to appear. · Zbl 0615.20040
[5] Petrich, M.,Lectures in Semigroups, Wiley, New York, 1977. · Zbl 0369.20036
[6] Scheiblich, H.E.,On coproducts of rectangular bands, Rocky Mountain J. Math. 8 (1978), 539–545. · Zbl 0391.20046 · doi:10.1216/RMJ-1978-8-3-539
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