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Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

2015
*
Journal of Algebra
*

Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are

doi:10.1016/j.jalgebra.2014.09.037
fatcat:sbstm56erfb27jjw3cxgqyvvai
*biautomatic*. ... For each a ∈ A ∪ {ε}, define the relations Hoffmann & Thomas have made a careful study of*biautomaticity*for*semigroups*[HT05] . ... They distinguish four notions of*biautomaticity*for*semigroups*: L a = {(u, v) : u, v ∈ L, ua = M v} a L = {(u, v) : u, v ∈ L, au = M v}. ...##
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Automaticity of One-Relator Semigroups with Length Less Than or Equal to Three

2017
*
Mathematics in Computer Science
*

Moreover, if u=v∈{ab=a, ab=b} then S is not

doi:10.1007/s11786-017-0291-7
fatcat:geicmg6y5jhlbo7kemdbbm2sxa
*biautomatic*. ... Such a*semigroup*is called a one-relator*semigroup*. Suppose that |v|≤|u|≤3, where |u| is the length of the word u. Suppose that a,b∈ A, a≠ b. ... If for any a ∈ A∪{ε}, a L $ , $ L a , a L $ and L $ a are all regular, then we say*semigroup*S has a*biautomatic*structure (A, L) and say that S is a*biautomatic**semigroup*. ...##
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Page 3683 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

subgroups of

*biautomatic*groups. ... In particular, the following properties are proved: Centralizers of*biautomatic*groups are*biautomatic*; a polycyclic subgroup of a*biautomatic*group is abelian by finite; Baumslag-Solitar groups are not ...##
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Decision problems for word-hyperbolic semigroups

2016
*
Journal of Algebra
*

It is proved that it is undecidable whether a word-hyperbolic

doi:10.1016/j.jalgebra.2016.07.007
fatcat:6opwwqw6qvg7xfqz6gerxxdg7e
*semigroup*is automatic, asynchronously automatic,*biautomatic*, or asynchronously*biautomatic*. ... (This does not alter the class of word-hyperbolic*semigroups*.) ... A monoid M is*biautomatic*if it Hoffmann & Thomas have made a careful study of*biautomaticity*for*semigroups*[18] . ...##
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Decision problems for word-hyperbolic semigroups
[article]

2015
*
arXiv
*
pre-print

It is proved that it is undecidable whether a word-hyperbolic

arXiv:1303.1763v2
fatcat:l7fjhpthcrgonj3si2jxyt2xvy
*semigroup*is automatic, asynchronously automatic,*biautomatic*, or asynchronously*biautomatic*. ... Algorithms are presented for deciding whether a word-hyperbolic*semigroup*is a monoid, a group, a completely simple*semigroup*, a Clifford*semigroup*, or a free*semigroup*. ... It is undecidable whether a word-hyperbolic*semigroup*is automatic (respectively, asynchronously automatic,*biautomatic*, asynchronously*biautomatic*). ...##
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The planar pure braid group is a diagram group
[article]

2021
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arXiv
*
pre-print

A number of consequences follow, including

arXiv:2109.02815v2
fatcat:yjmmuavmlvhlrny3f3wz7a2o7u
*biautomaticity*and bi-orderability of the groups Γ_n. Moreover, each group Γ_n acts properly and cocompactly on a CAT(0) cubical complex. ... It follows directly that each Γ n is linear, bi-orderable, and*biautomatic*. ... A result of Niblo and Reeves [17] shows that any group acting properly and cocompactly on a CAT(0) cubical complex is*biautomatic*. ...##
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On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

2017
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Information and Computation
*

The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being

doi:10.1016/j.ic.2017.05.003
fatcat:elvrvnoeczabfkiulmbp7gtywa
*biautomatic*are investigated for this class of monoids. ... The Hoffmann & Thomas have made a careful study of*biautomaticity*for*semigroups*[53] . ... One can ask: Does every homogeneous*semigroup*admit a presentation by a finite complete rewriting system? Is every such*semigroup**biautomatic*? ...##
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On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids
[article]

2017
*
arXiv
*
pre-print

The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being

arXiv:1407.7428v2
fatcat:fp7rno7blzewxfo5g5tsooiab4
*biautomatic*are investigated for this class of monoids. ... The Hoffmann & Thomas have made a careful study of*biautomaticity*for*semigroups*[53] . ... One can ask: Does every homogeneous*semigroup*admit a presentation by a finite complete rewriting system? Is every such*semigroup**biautomatic*? ...##
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Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids

2015
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International journal of algebra and computation
*

For hypoplactic monoids, we construct finite complete rewriting systems and

doi:10.1142/s0218196715400044
fatcat:3igjlhwmezh2rivpnnttcl6tiy
*biautomatic*structures. ... This paper studies complete rewriting systems and*biautomaticity*for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. ... For each a ∈ A ∪ {ε}, define the relations Hoffmann & Thomas have made a careful study of*biautomaticity*for*semigroups*[HT05] . ...##
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Finite transducers for divisibility monoids

2006
*
Theoretical Computer Science
*

In addition, we prove that every divisibility monoid is

doi:10.1016/j.tcs.2006.06.019
fatcat:k275oghjffhazovwhy3lvpr3ni
*biautomatic*. ... These four notions of automaticity are shown to be independent for general*semigroups*and to collapse into a dual notion of -automaticity for cancellative monoids (whether automaticity implies*biautomaticity*... Every left divisibility monoid is*biautomatic*. ...##
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Finite transducers for divisibility monoids
[article]

2006
*
arXiv
*
pre-print

In addition, we prove that every divisibility monoid is

arXiv:math/0601328v1
fatcat:ktrvqmn54nf37ptzmcn5idotxe
*biautomatic*. ... These four notions of automaticity are shown to be independent for general*semigroups*and to collapse into a dual notion of µautomaticity for cancellative monoids (whether automaticity implies*biautomaticity*...*Biautomaticity*The results from the previous section make us ready to establish the*biautomaticity*of left divisibility monoids. ...##
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Page 4159 of Mathematical Reviews Vol. , Issue 2003f
[page]

2003
*
Mathematical Reviews
*

Section 3 contains the proof of

*biautomaticity*. Right and left normal forms are defined, and the left form is shown to give the group a*biautomatic*structure. ... (English summary) International Conference on Geometric and Combinatorial Methods in Group Theory and*Semigroup*Theory (Lincoln, NE, 2000). Internat. J. Algebra Comput. 12 (2002), no. 1-2, 341-355. ...##
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Notions of hyperbolicity in monoids

2010
*
Theoretical Computer Science
*

Hyperbolic monoids (in the sense introduced here) turn out to be

doi:10.1016/j.tcs.2009.10.016
fatcat:g2vmhyxatbganjxnf7wk5hadey
*biautomatic*. ... A monoid M is said to be left-*biautomatic*if it has a left-*biautomatic*structure and right-*biautomatic*if it has a rightbiautomatic structure. Remark 6. ... It has been noted (see [15, 16] for example) that the definition of automaticity generalizes naturally from groups to*semigroups*and an exploration of the basic properties of automatic*semigroups*was ...##
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Page 4800 of Mathematical Reviews Vol. , Issue 93i
[page]

1993
*
Mathematical Reviews
*

ISBN 0-86720-244-0 Let G be a group; an automatic structure (A, L) on G consists of a set A of

*semigroup*generators of G, together with a language L accepted by a finite state automaton W over A, such ... Short) that a*biautomatic*group has solvable conjugacy problem, and the question is then posed as to whether or not an automatic group necessarily has solvable conjugacy problem. ...##
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On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

2011
*
Science China Mathematics
*

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for

doi:10.1007/s11425-011-4212-y
fatcat:oadxwmr52rebvjdfusyei2d3xi
*biautomatic*groups reduces to that for*biautomatic*groups ... Lee Mosher proved that central quotients of*biautomatic*groups are*biautomatic*[14] . ... We remind the reader that a rational structure for a group G with finite*semigroup*generators X is a regular language L ⊂ X * that maps bijectively to G under the natural map X * → G. Lemma 6.1. ...
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