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
Number of pages	32
Journal	Semigroup Forum
Volume	First Online
Early online date	14 Aug 2020
DOIs	https://doi.org/10.1007/s00233-020-10127-0
Publication status	E-pub ahead of print - 14 Aug 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
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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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T1 - On finitary properties for fiber products of free semigroups and free monoids

AU - Clayton, Ashley

PY - 2020/8/14

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N2 - 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.

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 - First Online

JO - Semigroup Forum

JF - Semigroup Forum

ER -

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

Abstract

Keywords

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