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Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple


We will study the presentations of fundamental groups of the complement of complexified real affine line arrangements that do not contain two parallel lines. By Yoshinaga’s minimal presentation, we can give positive homogeneous presentations of the fundamental groups. We consider the associated monoids defined by the presentations. It turns out that, in some cases, left (resp. right) least common multiple does not always exist. Hence, the monoids are neither Garside nor Artin. Nevertheless, we will show that they carry certain particular elements similar to the fundamental elements in Artin monoids and that, by improving the classical method in combinatorial group theory, they are cancellative monoids. As a result, we will show that the word problem can be solved and the center of them is determined.

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The author was very glad to participate in the conference JSPS-VAST Japan–Vietnam Bilateral Joint Projects “Topology of Singularities and Related Topics, III” Dalat, Vietnam, 26–30 March 2012. The author thanks all the organizers: Nguyen Viet Dung, Ta Le Loi, Pham Tien Son, Mutsuo Oka and Masaharu Ishikawa. The conference was supported by Japan Society for the Promotion of Science, Vietnamese Academy of Science and Technology, GDRI Singularities France–Japan–Vietnam (CNRS), National Foundation for Science Technology Development. The author is grateful to Kyoji Saito for very interesting discussions and encouragement. This research is supported by JSPS Fellowships for Young Scientists (23⋅10023). This research is also supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

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Correspondence to Tadashi Ishibe.

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Ishibe, T. Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple. Vietnam J. Math. 42, 305–326 (2014).

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  • Fundamental group
  • The word problem

Mathematics Subject Classification (2010)

  • 20F05