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The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts

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Abstract

In any lattice-ordered group (l-group) generated by a setX, every element can be written (not uniquely) in the form w(x)=⋁ i ⋀ j w ij (x), where eachw ij (x) is a group word in the elements ofX. An algorithm will be given for deciding whetherw(x) is the identitye in the free normal valuedl-group onX, or equivalently, whether the statement “∀x,w(x)=e” holds in all normal valuedl-groups. The algorithm is quite different from the one given recently by Holland and McCleary for the freel-group, and indeed the solvability of the word problem was established first for the normal valued case. The present algorithm makes crucial use of the fact (due to Glass, Holland, and McCleary) that the variety of normal valuedl-groups is generated by the finite wreath powersZ Wr Z Wr...Wr Z of the integersZ. In general, use of the algorithm requires a fairly large amount of work, but in several important special cases shortcuts are obtained which make the algorithm very quick.

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Authors and Affiliations

  1. University of Georgia, Athens, Geogia, USA

    Stephen H. McCleary

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  1. Stephen H. McCleary
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Additional information

This is an expanded version of material developed while the author was on leave at Bowling Green State University in Bowling Green, Ohio, and presented in 1978 at the Conference on Ordered Groups at Boise State University in Boise, Idaho [9].

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McCleary, S.H. The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts. Algebra Universalis 14, 317–348 (1982). https://doi.org/10.1007/BF02483936

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