International Journal of Algebra and Computation

We define the concept of an $L$-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for $L$-co-Hopfian groups is established as well as it is shown that there exists an $L$-co-Hopfian group which is not co-Hopfian.

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

For any integer $d\ge 1$, we construct examples of finitely presented associative algebras over a field of characteristic $\ne 2$ with intermediate growth of type $[e^{n^{d/(d+1)}}]$. We produce these examples by computing the growth types of some finitely presented metabelian Lie algebras.

We define a new invariant of a conjugacy class of subgroups which we call the breadth and prove that a quasiconvex subgroup of a negatively curved group has finite breadth in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex subgroup of a negatively curved group with its conjugate. Using that algorithm we construct algorithms for computing the breadth, the width, and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group. We also explicitly compute a quasiconvexity constant of the intersection of two quasiconvex subgroups and give examples demonstrating that height, width, and breadth are different invariants of a subgroup.

We develop new methods for computing the Hochschild (co)homology of monoids which can be presented as the structure monoids of idempotent set-theoretic solutions to the Yang–Baxter equation. These include free and symmetric monoids; factorizable monoids, for which we find a generalization of the Künneth formula for direct products; and plactic monoids. Our key result is an identification of the (co)homologies in question with those of the underlying YBE solutions, via the explicit quantum symmetrizer map. This partially answers questions of Farinati–García-Galofre and Dilian Yang. We also obtain new structural results on the (co)homology of general YBE solutions.