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Automorphisms over semigroups of matrices (two and higher order matrices)
Automorphisms over semigroups of matrices (two and higher order matrices)
The main motivation for us to come to this work was the correspondence and some intricate connections and differences, which exist between mathematics existing in to describe binary vs. non-binary mathematical structures.
In the paper [Bunina, Tupikina, 2013] we describe all automorphisms of semigroups of matrices of order k=2. However, the frameworks to describe all automorphisms of semigroups of such matrices, as well as all the automorphisms of matrices themselves are very different when it comes to matrices of order k=3 and kxk, k>3. There are some specific reasons for this and we are going to describe it soon below. What we are interested here are the core differences in treating automorphisms for semigroups of matrices of the sizes 2 or higher than 2.
In [Bunina, Tupikina, 2013] we have considered and proved properties of all automorphisms of semigroups of matrices over associative but not neccerily commutative rings for the special case of matrix size k=2 (different from the case k=3). Now as an extension we want to consider and find the general properties of all automorphisms of semigroups of matrices over not neccesarily associative rings R. The main idea would be first to try to apply it to the centre of such non-associative ring R, and then extend it to the matrices which commute with matrices in the centre.
Definition. Centre of algebra (semigroup) A.
References:
[Bunina, Tupikina, 2013] https://link.springer.com/article/10.1007/s10958-012-0815-2 and https://www.mathnet.ru/php/getFT.phtml?jrnid=fpm&paperid=1364&what=fullt&option_lang=eng
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