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From A to B to Z

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posted on 2025-04-01, 04:11 authored by Marcel JacksonMarcel Jackson, WT Zhang
The variety generated by the Brandt semigroup B2 can be defined within the variety generated by the semigroup A2 by the single identity x2y2≈ y2x2. Edmond Lee asked whether or not the same is true for the monoids B21 and A21. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of A21 that satisfy x2y2≈ y2x2 and contain B21. A further consequence is that the variety of B21 cannot be defined within the variety of A21 by any finite system of identities. Continuing downward, we then turn to subvarieties of B21. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity x2y≈ yx2 and containing the monoid M(z∞) , where z∞ denotes the infinite limit of the Zimin words z= x, zn+1= znxn+1zn.

Funding

Marcel Jackson was supported by ARC Future Fellowship FT120100666 and Wen Ting Zhang by ARC Discovery Project DP1094578, the National Natural Science Foundation of China (Nos. 11771191, 11401275, 11371177) and the Natural Science Foundation of Gansu Province (No. 20JR5RA275).

History

Publication Date

2021-08-01

Journal

Semigroup Forum

Volume

103

Issue

1

Pagination

26p. (p. 165-190)

Publisher

Springer

ISSN

0037-1912

Rights Statement

© 2021 The Authors. This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use (see https://www.springernature.com/gp/open-research/policies/accepted-manuscript-terms), but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00233-021-10180-3

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