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Monoids and Groups of I-Type

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Abstract

A monoid S generated by {x 1,. . .,x n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaMn of rank n generated by {u 1,. . .,u n} to S so that for all a∈FaMn one has {v(u 1 a),. . .,v(u n a)}={x 1 v(a),. . .,x n v(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh.

In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fan and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups.

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Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, Belgium

    Eric Jespers

  2. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Jan Okniński

Authors
  1. Eric Jespers
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  2. Jan Okniński
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Corresponding author

Correspondence to Eric Jespers.

Additional information

In memory of Paul Wauters

Mathematics Subject Classifications (2000)

20F05, 20M05; 16S34, 16S36, 20F16.

The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).

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Jespers, E., Okniński, J. Monoids and Groups of I-Type. Algebr Represent Theor 8, 709–729 (2005). https://doi.org/10.1007/s10468-005-0342-7

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