Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q)$ is indecomposable, with the exception of one case, where two indecomposable components arise.Comment: 27 page

Invertible and nilpotent matrices over antirings

In this paper we characterize invertible matrices over an arbitrary commutative antiring S and find the structure of GL_n (S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent $n \times n$ matrix over an entire antiring can be written as a sum of $\lceil \log_2 n \rceil$ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.Comment: 9 pages, 1 figure, minor change

The unitary Cayley graph of a semiring

We study the unitary Cayley graph of a matrix semiring. We find bounds for its diameter, clique number and independence number, and determine its girth. We also find the relationship between the diameter and the clique number of a unitary Cayley graph of a semiring $S$ and a matrix semiring over $S$

Nilpotent matrices over antirings

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\ud 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