Abstract
Marek Kuczma asked in 1980 whether for every positive integer n, there exists a subsemigroup M of a group G, such that G is equal to the n-fold product \(M\,M^{-1} M\,M^{-1} \ldots \,M^{(-1)^{n-1}}\), but not to any proper initial subproduct of this product. We answer his question affirmatively, and prove a more general result on representing a certain sort of relation algebra by a family of subsets of a group. We also sketch several variants of the latter result.
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Presented by J.B. Nation.
To the memory of Bjarni Jónsson.
This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.
Work partly supported by NSF Contract MCS 80-02317. (The author apologizes for the long delay in publishing this work, most of which was done in 1981.)
http://arxiv.org/abs/1702.06088. After publication of this note, updates, errata, related references etc., if found, will be recorded at http://math.berkeley.edu/~gbergman/papers/.
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Bergman, G.M. Submonoids of groups, and group-representability of restricted relation algebras. Algebra Univers. 79, 19 (2018). https://doi.org/10.1007/s00012-018-0488-x
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DOI: https://doi.org/10.1007/s00012-018-0488-x