scholarly journals Factoriality and the Pin-Reutenauer procedure

2016 ◽  
Vol Vol. 18 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
J. Almeida ◽  
J. C. Costa ◽  
M. Zeitoun
Keyword(s):  
Free Group ◽  
Free Algebra ◽  
Natural Analogue ◽  
Finitely Generated ◽  
Finite Semigroups ◽  
Rational Language ◽  
Profinite Topology

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

1991 ◽  
Vol 01 (04) ◽  
pp. 411-436 ◽  
Author(s):  
KARSTEN HENCKELL ◽  
STUART W. MARGOLIS ◽  
JEAN-ERIC PIN ◽  
JOHN RHODES
Keyword(s):  
Free Group ◽  
Semigroup Theory ◽  
Direct Proof ◽  
Type Ii ◽  
Finitely Generated ◽  
Free Monoids ◽  
Finite Monoids ◽  
Rational Subset ◽  
Block Groups ◽  
Profinite Topology

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture — also verified by Ash — it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2…Hn, where g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of M. Hall. Finally, we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.

FREE PRODUCT, PROFINITE TOPOLOGY AND FINITELY GENERATED SUBGROUPS

2001 ◽  
Vol 11 (02) ◽  
pp. 171-184 ◽  
Author(s):  
THIERRY COULBOIS
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Groups ◽  
The Other ◽  
Finitely Generated ◽  
Product Operation ◽  
Other Hand ◽  
The One ◽  
Profinite Topology

We consider the following property for a group G:(RZn)ifH1,…,Hnare finitely generated subgroups of G then the setH1 H2⋯ Hn= {h1 ⋯ hn| h1∈ H1, …,hn∈ Hn}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property ( RZ n). L. Ribes and P. Zalesskiĭ proved that any free group has ( RZ n). We show that the property ( RZ n) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

Finitely generated free Heyting algebras

1986 ◽  
Vol 51 (1) ◽  
pp. 152-165 ◽  
Author(s):  
Fabio Bellissima
Keyword(s):  
Free Algebra ◽  
Kripke Semantics ◽  
Heyting Algebras ◽  
Free Algebras ◽  
Finitely Generated ◽  
Algebraic Properties

AbstractThe aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

1979 ◽  
Vol 31 (6) ◽  
pp. 1329-1338 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Products

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

Sign in / Sign up

Export Citation Format

Share Document