Nilpotent matrices over antirings

Klarisa Trojer (2018) Nilpotent matrices over antirings. EngD thesis.

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Abstract

The main goal of this thesis is to decribe the properties of nilpotent matrices over antirings. We start with the introduction of the abstract structures, more precisely, groups and semirings. We observe the properties that apply to semigroups, monoids and groups, and also the properties that apply to semirings. After a brief description of the history of previous studies done on matrices over semirings, we give some definitions and lemmas that are used later on. We define what an antiring is and talk about nilpotence of elements and matrices, we also define the permanent and the associated permanent minors. Next, we define and describe the properties of the nilpotent matrices. We also write about simultaneous nilpotency and the nilpotent index. At the end we provide a method for calculationg the nilpotent index.

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