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Doctoral Thesis
Full name
Giliard Souza dos Anjos
Knowledge Area
Date of Defense
São Paulo, 2018
Giuliani, Maria de Lourdes Merlini (President)
Barros, Dylene Agda Souza de
Grichkov, Alexandre
Guzzo Junior, Henrique
Kochloukov, Plamen Emilov
Title in Portuguese
Half-Isomorfismos de loops automórficos
Keywords in Portuguese
Anel de Lie
Grupo de half-automorfismos
Loop automórfico
Loop automórfico de Lie
Loop automórfico diedral
Abstract in Portuguese
Loops automórficos, ou A-loops, são loops nos quais todas as aplicações internas são automorfismos. Esta variedade de loops inclui grupos e loops de Moufang comutativos. Loops automórficos diedrais formam uma classe de A-loops construda a partir da duplicação de grupos abelianos finitos, generalizando a construção do grupo diedral. Outra classe de A-loops é a dos loops automórficos de Lie, construda a partir de anéis de Lie, definindo-se uma nova operação entre seus elementos. Um half-isomorfismo é uma bijeção f entre loops L e L' onde, para quaisquer x e y pertencentes a L, temos que f(xy) pertence ao conjunto {f(x)f(y),f(y)f(x)}. Dizemos que o half-isomorfismo f é não trivial quando f não é um isomorfismo e nem um anti-isomorfismo. Nesta tese descrevemos propriedades de half-isomorfismos de loops, classificamos os half-isomorfismos entre loops automórficos diedrais e obtivemos o grupo de half-automorfismos nesta classe. Para os loops automórficos de Lie de ordem mpar, mostramos que todo half-automorfismo é trivial.
Title in English
Half-isomorphisms of automorphic loops
Keywords in English
Automorphic loop
Dihedral automorphic loop
Group of half-automorphisms
Lie automorphic loop
Lie ring
Abstract in English
Automorphic loops, or A-loops, are loops in which every inner mapping is an automorphism. This variety of loops includes groups and commutative Moufang loops. Dihedral automorphic loops form a class of A-loops, constructed from the duplication of finite abelian groups, that generalizes the construction of the dihedral group. Another class of A-loops is the Lie automorphic loops, constructed from Lie rings, where a new operation between its elements is defined. A half-isomorphism is a bijection f between loops L and L' where, for any x and y belong to L, we have that f(xy) belongs to the set {f(x)f(y),f(y)f(x)}. We say that half-isomorphism f is non trivial when f is neither an isomorphism nor an anti-isomorphism. In this thesis, we describe properties of half-isomorphisms of loops, we classify the half-isomorphisms between dihedral automorphic loops and we obtain the group of half-automorphisms in this class. For the Lie automorphic loops of odd order, we show that every half-automorphism is trivial.
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