Context-free rewriting systems and word-hyperbolic structures with uniqueness

This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan & Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid)

Markov semigroups, monoids, and groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.Comment: 40 pages; 3 figure

Uniform decision problems in automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse.Comment: 19 page

Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2

The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time