arXiv subjects

Giliard Souza dos Anjos

Publications and source records attributed to Giliard Souza dos Anjos.

6 recordsLinked to original sources

arXiv2022

Half-isomorphisms of automorphic loops

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a bijection from $G$ onto $K$ such that $f(ab)\in\{f(a)f(b), f(b)f(a)\}$ for any $a,b\in G$. A half-isomorphism is trivial when it is either an isomorphism or an anti-isomorphism. Consider the class of automorphic loops such that the equation $x\cdot(x\cdot y) = (y\cdot x)\cdot x $ is equivalent to $x\cdot y = y\cdot x$. Here we show that this class of loops includes automorphic loops of odd order and uniquely $2$-divisible. Furthermore, we prove that every half-isomorphism between loops in that class is trivial.

math.GR↗

arXiv2022

Free Bol loops of exponent two

A Bol loop is a loop that satisfies the identity $x((yz)y)=((xy)z)y$. In this paper, we give a construction of the free Bol loops of exponent two. We define a canonical form of all their elements and describe their multiplication law based on this form.

math.GR↗

arXiv2022

Half-automorphism group of a class of Bol loops

A Bol loop is a loop that satisfies the Bol identity $(xy.z)y=x(yz.y)$. If $L$ is a loop and $f:L\to L$ is a bijection such that $f(xy)\in\{f(x)f(y),f(y)f(x)\}$, for every $x$, $y\in L$, then $f$ is called a half-automorphism of $L$. In this paper, we describe the half-automorphism group of a class of Bol loops of order $4m$.

math.GR↗

arXiv2020

Half-automorphism group of Chein loops

A bijection $f$ of a loop $L$ is a half-automorphism if $f(xy)\in \{f(x)f(y),f(y)f(x)\}$, for any $x,y\in L$. A half-automorphism is nontrivial when it is neither an automorphism nor an anti-automorphism. A Chein loop $L=G\cup Gu$ is a Moufang loop constructed from a group $G$ and an element $u$ of order $2$ outside of $G$. In this paper, the half-automorphism group of finite Chein loops is described.

math.GR↗

arXiv2020

Half-isomorphisms whose inverses are also half-isomorphisms

Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a half-isomorphism if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$, for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism $f$ is called special if $f^{-1}$ is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.

math.GR↗

arXiv2015

Steiner Loops Satisfying the Statement of Moufang's Theorem

Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem. We give a partial answer to this question and present two types of Steiner loops, one that satisfies Moufang's theorem and another that does not, and neither is Moufang loop.

math.GR↗