In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y, f) is a finitely presented system, then there exists a Smale space (X,F ) and a u-resolving factor map π+ : X → Y . If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we s...

A nilpotent quotient algorithm for finitely presented Lie rings over Z (LIENQ) is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A nilpotent presentation consists of generators for the abelian group and the products expressed as linear combinations for pairs formed...

Given a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quas...

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidabl...

We explore natural classes of finitely generated semigroups that have word problem decidable by synchronous or asynchronous two-tape finite state automata. Synchronous two-tape automata decide regular word problems and asynchronous automata decide rational word problems. We argue that asynchronous two-tape automata are a more suitable choice than synchronous two-tape automata and show that the ...

Given a finite presentation of a group G, proving properties of G can be difficult. Indeed, many questions about finitely presented groups are unsolvable in general. Algorithms exist for answering some questions while for other questions algorithms exist for verifying the truth of positive answers. An important tool in this regard is the Todd-Coxeter coset enumeration procedure. It is possible ...

A group G given by a presentation G = 〈A ‖ R〉 is called weakly finitely presented if every finitely generated subgroup of G, generated by (images of) some words in A, is naturally isomorphic to the subgroup of a group G0 = 〈A0 ‖ R0〉, where A0 ⊆ A, R0 ⊆ R are finite, generated by (images of) the same words. In the article, weakly finitely presented periodic groups which are not locally finite ar...

The filling radius function R of Gromov measures the minimal radii of van Kampen diagrams filling edge-circuits w in the Cayley 2complex of a finite presentation P. It is known that the Dehn function can be bounded above by a double exponential in R and the length of the loop, and it is an open question whether a single exponential bound suffices. We define the upper filling radius R(w) of w to...

Let L be a free Lie algebra over a field k, I a non-trivial proper ideal of L, n > 1 an integer. The multiplicator H2(L/I , k) of L/I is not finitely generated, and so in particular, L/I is not finitely presented, even when L/I is finite dimensional.