On the decidability of the word problem for amalgamated free products of inverse semigroups

We study inverse semigroup amalgams [S ₁,S ₂;U], where S ₁ and S ₂ are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S ₁ and S ₂. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S ₁* _U S ₂ to state sufficient conditions for the amalgamated free products S ₁* _U S ₂ having decidable word problem.     

Alternating subgroups of Coxeter groups

We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.     

The next step of the word problem over monoids

It is known that a group presentation P can be regarded as a 2-complex with a single 0-cell. Thus we can consider a 3-complex with a single 0-cell which is known as a 3-presentation. Similarly, we can also consider 3-presentations for monoids. In this paper, by using spherical monoid pictures, we show that there exists a finite 3-monoid-presentation which has unsolvable “generalized identity problem” that can be thought as the next step (or one-dimension higher) of the word problem for monoids. We note that the method used in this paper has chemical and physical applications.     

On Presentations of Bruck–Reilly Extensions

We first consider the class of monoids in which every left invertible element is also right invertible, and prove that if a monoid belonging to this class admits a finitely presented Bruck–Reilly extension then it is finitely generated. This allow us to obtain necessary and sufficient conditions for the Bruck–Reilly extensions of this class of monoids to be finitely presented. We then prove that thes 𝒟-classes of a Bruck–Reilly extension of a Clifford monoid are Bruck–Reilly extensions of groups. This yields another necessary and sufficient condition for these Bruck–Reilly extensions to be finitely generated and presented. Finally, we show that a Bruck–Reilly extension of a Clifford monoid is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid, and that this property cannot be generalized to Bruck–Reilly extensions of arbitrary inverse monoids.     

Bruhat Presentations for ∗-Classical Groups

We introduce in a general setting, the unitary similitude groups associated to ε-hermitian form over a ring A with involution. We concentrate next in the rank 2 case. The classical presentation of GL(2, F), F a field, originating from its Bruhat decomposition can be extended to these unitary similitude groups. Furthermore, the multiplier associated to ε-hermitian forms affords a signed non commutative version of the classical determinant of 2 by 2 matrices, which we recover exploiting Grassmann's approach to determinants. Finally, we present various examples of our groups and give some categorical properties of them.     

On the Existence of 3-Generator, 3-Relation Finite 2-Groups with a Given (Co)Class

In this article, we show that for any positive integer k there is a 3-generator, 3-relation finite 2-group of class (respectively, coclass) k provided that k ≥ 4 (respectively, k ≥ 5).     

Filtrations of Totally Reflexive Modules

In this paper, we introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper triangular presentation matrix and that such a module has a complete resolution in which every differential can be simultaneously represented by upper triangular matrices.     

Rewriting systems in alternating knot groups with the Dehn presentation

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating.      

Groups that are Pairwise Nilpotent

In this article we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.