We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an i...

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following ...

If V is a commutative algebraic group over a field k, O is a commutative ring that acts on V , and I is a finitely generated free O-module with a right action of the absolute Galois group of k, then there is a commutative algebraic group I ⊗O V over k, which is a twist of a power of V . These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori...

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups that admit a descending chain of normal finite-index subgroups, each which G. prove up finite index, these are always obtained by pulling back from free abelian quotient. give two applications: First, we show characteristic subgroup arises the ab...

The concept of an automatic structure has been generalized from groups [ECH+] to semigroups [CRRT]. Several authors [CRR, Hof, Kam] have asked the following question: Let S be a finitely generated semigroup embeddable in a group and let G be its universal group [CP, Chapter ]. If G is automatic, must S be automatic? Examples in favour of this implication include: free groups and s...

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that i...

let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

It has been asserted that any (full) order on a torsion-free, finitely generated, nilpotent group is defined by some F-basis of G and that the group of o-automorphisms of such a group is itself a group of the same kind: Examples provided herein demonstrate that both of these assertions are false; however, it is proved that the group of o-automorphisms of an ordered, polycyclic group is nilpoten...

Let K be a locally finitely presentable category. If K is abelian and the sequence 0 K // X // k // C c // // 0 // is short exact, we show that 1) K is finitely generated⇔ c is finitely presentable; 2) k is finitely presentable⇔ C is finitely presentable. The “⇐” directions fail for semi-abelian varieties. We show that all but (possibly) 2)(⇐) follow from analogous properties which hold in all ...