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

Alan J. Cain, Robert D. Gray, António Malheiro
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 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}.  ...

### Automaticity of One-Relator Semigroups with Length Less Than or Equal to Three

Yuqun Chen, Haibin Wu, Honglian Xie
2017 Mathematics in Computer Science
Moreover, if u=v∈{ab=a, ab=b} then S is not 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.  ...

### 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  ...

### Decision problems for word-hyperbolic semigroups

Alan J. Cain, Markus Pfeiffer
2016 Journal of Algebra
It is proved that it is undecidable whether a word-hyperbolic 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] .  ...

### Decision problems for word-hyperbolic semigroups [article]

Alan J. Cain, Markus Pfeiffer
2015 arXiv   pre-print
It is proved that it is undecidable whether a word-hyperbolic 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).  ...

### The planar pure braid group is a diagram group [article]

Daniel S. Farley
2021 arXiv   pre-print
A number of consequences follow, including 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.  ...

### On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

Alan J. Cain, Robert D. Gray, António Malheiro
2017 Information and Computation
The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being 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?  ...

### On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids [article]

Alan J. Cain, Robert Gray, António Malheiro
2017 arXiv   pre-print
The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being 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?  ...

### Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids

Alan J. Cain, Robert D. Gray, António Malheiro
2015 International journal of algebra and computation
For hypoplactic monoids, we construct finite complete rewriting systems and 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] .  ...

### Finite transducers for divisibility monoids

Matthieu Picantin
2006 Theoretical Computer Science
In addition, we prove that every divisibility monoid is 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.  ...

### Finite transducers for divisibility monoids [article]

Matthieu Picantin
2006 arXiv   pre-print
In addition, we prove that every divisibility monoid is 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.  ...

### 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.  ...

### Notions of hyperbolicity in monoids

Michael Hoffmann, Richard M. Thomas
2010 Theoretical Computer Science
Hyperbolic monoids (in the sense introduced here) turn out to be 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  ...

### 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.  ...

### On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

Martin R. Bridson, Lawrence Reeves
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 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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