Collatz Numbers
In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$.
math.GM↗
arXiv subjects
A. C. Souza Filho
Publications and source records attributed to A. C. Souza Filho.
Traces of Torsion Units
A conjecture due to Zassenhaus asserts that if $\ G$ is a finite group then any torsion unit in $\mathbb{Z}G$ is conjugate in $\mathbb{Q}G$ to an element of $\ G$. We present a weaker form of this conjecture for some infinite groups.
math.GR↗
Alternative algebras with the hyperbolic property
We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $Γ\subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units of $\U (Γ)$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem.
math.GR↗
Free Groups in Quaternion Algebras
In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over $\Q (\sqrt{-d})$, $d \equiv 7 \pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\in \mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\ \mathbb{Q}$, thus including matrix algebras. More precisely, we show that $< u^n,v^n> $ is a free group for all $n\geq 1$ and $d>2$ and for $d=2$ and all $n\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free.
math.GR↗
Hyperbolic Unit Groups and Quaternion Algebras
We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R.
math.RA↗
Hyperbolicity of Semigroup Algebras II
In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra %over a field of characteristic zero, contains a ${\mathbb{Z}}$-order with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semigroup is semi-simple.
math.RA↗
Hyperbolicity of Semigroup Algebras
Let $A$ be a finite dimensional $Q-$algebra and $Γsubset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(Γ)$. We call this last property the hyperbolic property. We apply this in the case that $A = KS$ a semigroup algebra with $K = Q$ or $K = Q(\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n \in \{5,8,12\}$.
math.RA↗
On the Classification of Finite Semigroups and RA-loop with the Hyperbolic Property
We classify the finite semigroups S, for which all the Z-orders O of the rational Q-algebra QS, is such that the unit group U(O) is hyperbolic. We also classify the RA-loops L, for which the unit loop U(ZL) does not contain any free abelian subgroup of rank two.
math.RA↗