Generators and presentations for direct and wreath products of monoid acts

Craig Miller

doi:10.1007/s00233-018-9987-5

Generators and presentations for direct and wreath products of monoid acts

Craig Miller

Pure Mathematics

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Abstract

We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property P in direct products if, for any two M-acts A and B, the direct product A x B has property P if and only if both A and B have property P. It is proved that the monoids M that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal M-act M x M is finitely generated (resp. finitely presented). We show that a wreath product A ≀ B is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for A ≀ B to be finitely presented is that both A and B are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented.

Original language	English
Number of pages	24
Journal	Semigroup Forum
Volume	First Online
Early online date	17 Dec 2018
DOIs	https://doi.org/10.1007/s00233-018-9987-5
Publication status	E-pub ahead of print - 17 Dec 2018

Keywords

Monoid act
Presentation
Direct product
Wreath product

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Miller2018_Article_GeneratorsAndPresentationsForD
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Cite this

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title = "Generators and presentations for direct and wreath products of monoid acts",

abstract = "We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property P in direct products if, for any two M-acts A and B, the direct product A x B has property P if and only if both A and B have property P. It is proved that the monoids M that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal M-act M x M is finitely generated (resp. finitely presented). We show that a wreath product A ≀ B is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for A ≀ B to be finitely presented is that both A and B are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented.",

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author = "Craig Miller",

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TY - JOUR

T1 - Generators and presentations for direct and wreath products of monoid acts

AU - Miller, Craig

N1 - The author would like to thank the EPSRC for financial support.

PY - 2018/12/17

Y1 - 2018/12/17

N2 - We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property P in direct products if, for any two M-acts A and B, the direct product A x B has property P if and only if both A and B have property P. It is proved that the monoids M that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal M-act M x M is finitely generated (resp. finitely presented). We show that a wreath product A ≀ B is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for A ≀ B to be finitely presented is that both A and B are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented.

AB - We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property P in direct products if, for any two M-acts A and B, the direct product A x B has property P if and only if both A and B have property P. It is proved that the monoids M that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal M-act M x M is finitely generated (resp. finitely presented). We show that a wreath product A ≀ B is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for A ≀ B to be finitely presented is that both A and B are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented.

KW - Monoid act

KW - Presentation

KW - Direct product

KW - Wreath product

U2 - 10.1007/s00233-018-9987-5

DO - 10.1007/s00233-018-9987-5

M3 - Article

SN - 0037-1912

VL - First Online

JO - Semigroup Forum

JF - Semigroup Forum

ER -

Generators and presentations for direct and wreath products of monoid acts

Abstract

Keywords

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