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

Ashley Clayton

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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.
Original languageEnglish
Number of pages32
JournalSemigroup Forum
VolumeFirst Online
Early online date14 Aug 2020
DOIs
Publication statusE-pub ahead of print - 14 Aug 2020

Keywords

  • Subdirect product
  • Fiber product
  • Semigroup
  • Free semigroup
  • Free monoid

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