Linear preservers of term ranks of matrices over semirings

The term rank of a matrix $A$ over a semiring $\mathbf{S}$ is the least number of lines (rows or columns) needed to include all the nonzero entries in $A$. In this paper, we study linear operators that preserve term ranks of matrices over $\mathbf{S}$. In particular, we show that a linear operator $T$ on matrix space over $\mathbf{S}$ preserves term rank if and only if $T$ preserves term ranks $1$ and $\alpha (\ge 2)$ if and only if $T$ preserves two consecutive term ranks in a restricted condition. Other characterizations of term-rank preservers are also given.     

Linear mappings preserving coincidence of factorized and boundary ranks of matrices over semirings

A problem of characterization of linear bijective mappings preserving coincidence of factorized and boundary ranks over a semiring is considered. A complete classification of those mappings over antinegative commutative rings with a unit and without divisors of zero is obtained.     

Hidden matrix semirings

Criteria analogous to Robson’s and Fuchs’ are given for a semiring to be isomorphic to a full matrix semiring. The necessity of additional conditions (compared with the case of rings) is investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 13–19, 2003.     

Extreme preservers of maximal column rank inequalities of matrix sums over semirings

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme properties with respect to maximal column rank inequalities of matrix sums over semirings. This work was supported by the research grant of the Cheju National University in 2007.     

Convolution algebras over complete semirings

Constructions of complete semirings via categorical algebra are provided, that is, we construct convolution algebras of given complete semirings via so-called action networks (i.e., small covariant categories). We aim to get some structural insight into the principle of unrestricted convolution.     

Primitive matrices over polynomial semirings

An extension of the definition of primitivity of a real nonnegative matrix to matrices with univariate polynomial entries is presented. Based on a suitably adapted notion of irreducibility an analogue of the classical characterization of real nonnegative primitive matrices by irreducibility and aperiodicity for matrices with univariate polynomial entries is given. In particular, univariate polynomials with nonnegative coefficients which admit a power with strictly positive coefficients are characterized. Moreover, a primitivity criterion based on almost linear periodic matrices over dioids is presented.     

Determinants of matrices over semirings

In this paper, the concept of determinants for the matrices over a commutative semiring is introduced, and a development of determinantal identities is presented. This includes a generalization of the Laplace and Binet–Cauchy Theorems, as well as on adjoint matrices. Also, the determinants and the adjoint matrices over a commutative difference-ordered semiring are discussed and some inequalities for the determinants and for the adjoint matrices are obtained. The main results in this paper generalize the corresponding results for matrices over commutative rings, for fuzzy matrices, for lattice matrices and for incline matrices.     

Two concepts of singularity for matrices over semirings

The two common concepts of singularity for matrices over semirings are being studied since the 1970’s and arise from natural generalizations of the determinant and linear dependence. They were introduced in the context of schedule algebras by Gondran and Minoux, who proved later that the concepts discussed are equivalent over any selective invertible semiring. We present an approach that uses a generalization of power series arithmetic and, in particular, allows to derive a short proof for the theorem of Gondran and Minoux. Our main result is a complete concise characterization of semirings over which the two concepts of singularity are equivalent.     

Commuting graphs of matrices over semirings

We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.     

Noncommuting graphs of matrices over semirings

We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2×2 nilpotent matrices.     

Regular matrices and their strong preservers over semirings

An m×n matrix A over a semiring is called regular if there is an n×m matrix G over such that AGA=A. We study the problem of characterizing those linear operators T on the matrices over a semiring such that T(X) is regular if and only if X is. Complete characterizations are obtained for many semirings including the Boolean algebra, the nonnegative reals, the nonnegative integers and the fuzzy scalars.     

Bases in semilinear spaces over zerosumfree semirings

This paper investigates the cardinality of a basis and the characterizations of a basis in semilinear space of n-dimensional vectors over zerosumfree semirings. First, it discusses the cardinality of a basis and gives some necessary and sufficient conditions that each basis has the same number of elements. Then it presents some conditions that a set of vectors is a basis and that a set of linearly independent vectors can be extended to a basis. In the end, it shows a necessary and sufficient condition that two semilinear spaces are isomorphic.