On finitary properties for fiber products of free semigroups and free monoids
- PDF / 2,933,943 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 99 Downloads / 165 Views
On finitary properties for fiber products of free semigroups and free monoids Ashley Clayton1 Received: 25 June 2019 / Accepted: 1 July 2020 © The Author(s) 2020
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. Keywords Subdirect product · Fiber product · Semigroup · Free semigroup · Free monoid Mathematics Subject Classification Primary: 20MO5 · Secondary: 08B26
1 Introduction A subdirect product of two algebras A and B is a subalgebra of the direct product, for which the natural projections onto A and B are surjective. In particular, the direct product of two algebras is a subdirect product, for which finitary properties have been well studied for groups. Most results indicate that direct products of groups have a well behaved structure based on their constituent factors. That is, Communicated by Victoria Gould. * Ashley Clayton ac323@st‑andrews.ac.uk 1
School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK
13
Vol.:(0123456789)
A. Clayton
two groups G and H have the following properties (amongst others) if and only if G × H also does: finitely generated; finitely presented; residually finite; nilpotent; solvable; and having decidable word problem. By way of contrast, subdirect products of groups have more complicated behaviour in general, which has been particularly well exhibited for subdirect products of free groups. There are examples (stemming from [1, Theorem 1]) which are not finitely generated [2, Example 3]; finitely generated without being finitely presented [6]; and finitely generated but with undecidable membership problem [10]. Describing their substructure complexity, any two non-abelian free groups G and H have uncountably many pairwise non-isomorphic subdirect products of G and H [2, Corollary B]. By a result due to Goursat [5], subdirect products of groups arise as fiber products and vice versa, and are hence constructible in some sense. By a comparatively more recent result due to Fleischer [3], this is also true more generally for varieties of algebras which are congruence permutable (that is, all congruences commute with each other under compositio
Data Loading...
