On Cosets in Split Extensions of Hypercomplex Numbers

Abstract

This work focuses on the study of some properties of cosets in split extensions of hypercomplex numbers. It is well known that if G is a group and H its subgroup, the cosets of the subgroup H form a partition of the group G. However, this property does not generally hold for loops. This study aims at constructing cyclic subloops of the split extensions of hypercomplex numbers and the corresponding cosets arising from them. It is then shown that the cosets of a cyclic subloop form a partition of the split extension loop i.e. any two right or left cosets of a cyclic subloop are either disjoint or identical. The study uses the Cayley-Dickson and Jonathan Smith doubling processes to construct multiplication tables for the split extensions of hypercomplex numbers. Nim addition is also used to give a general way of generating cyclic subloops and the cosets arising from them. In Loop Theory, only when 𝑆 is a normal subloop of 𝐿 will the left and right cosets of 𝑆 coincide, these cosets form a loop 𝐿/𝑆 called the quotient or factor loop whose multiplication is defined by (𝑎∙𝑆)∙(𝑏∙𝑆)=(𝑎∙𝑏)∙𝑆, ∀ 𝑎,𝑏∈𝐿. In this work we use cyclic normal subloops of split extensions of hypercomplex numbers to construct quotient loops. Lastly, we show that the multiplication of the elements in the quotient loop formed can also be carried out by considering the Nim addition of the subscripts of the individual elements.