On finitary properties for fiber products of free semigroups and free monoids

Ashley Clayton

doi:10.1007/s00233-020-10127-0

On finitary properties for fiber products of free semigroups and free monoids

Ashley Clayton

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and J-trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

Original language	English
Pages (from-to)	326-357
Number of pages	32
Journal	Semigroup Forum
Volume	101
Issue number	2
Early online date	14 Aug 2020
DOIs	https://doi.org/10.1007/s00233-020-10127-0
Publication status	Published - 1 Oct 2020

Keywords

Subdirect product
Fiber product
Semigroup
Free semigroup
Free monoid

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10.1007/s00233-020-10127-0Licence: CC BY

Clayton_2020_SI_Fiberproducts_CC
Copyright © The Author(s) 2020. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Final published version, 2.11 MBLicence: CC BY

Subdirect products of free semigroups and monoids
Clayton, A. (Author), Ruskuc, N. (Supervisor) & Mitchell, J. (Supervisor), 1 Dec 2020
Student thesis: Doctoral Thesis (PhD)

Cite this

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abstract = "We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and J-trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.",

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AB - We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and J-trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

KW - Subdirect product

KW - Fiber product

KW - Semigroup

KW - Free semigroup

KW - Free monoid

U2 - 10.1007/s00233-020-10127-0

DO - 10.1007/s00233-020-10127-0

M3 - Article

SN - 0037-1912

VL - 101

SP - 326

EP - 357

JO - Semigroup Forum

JF - Semigroup Forum

IS - 2

ER -

On finitary properties for fiber products of free semigroups and free monoids

Abstract

Keywords

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Student theses

Subdirect products of free semigroups and monoids

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