{
  "id": "1802.02420",
  "version": "v1",
  "published": "2018-02-07T13:49:40.000Z",
  "updated": "2018-02-07T13:49:40.000Z",
  "title": "A group-theoretical interpretation of the word problem for free idempotent generated semigroups",
  "authors": [
    "Yang Dandan",
    "Igor Dolinka",
    "Victoria Gould"
  ],
  "comment": "35 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\\mathsf{IG}(\\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\\mathcal{E}$ of idempotents. We show that when $\\mathcal{E}$ is finite, the word problem for $\\mathsf{IG}(\\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of $\\mathsf{IG}(\\mathcal{E})$. As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite $\\mathcal{E}$, $\\mathsf{IG}(\\mathcal{E})$ is always a weakly abundant semigroup satisfying the congruence condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-07T13:49:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M05",
      "20F10",
      "68Q70"
    ],
    "keywords": [
      "free idempotent generated semigroup",
      "word problem",
      "group-theoretical interpretation",
      "constraint satisfaction problems",
      "biordered set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}