Seok-Zun Song

Linear operators that preserve pairs of matrices which satisfy extreme rank properties

Abstract A pair of m × n matrices ( A , B ) is called rank-sum-maximal if rank( A + B )=rank( A )+rank( B ), and rank-sum-minimal if rank( A + B )=|rank( A )−rank( B )|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min( m , n )+2 elements and of characteristic not 2.

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.

Minimum permanents on certain faces of matrices containing an identity submatrix

Abstract We determine the minimum permanents on certain faces of Ω n for the fully indecomposable (0, 1) matrices containing an identity submatrix of some order. We also determine whether the given fully indecomposable (0, 1) matrices are either cohesive and barycentric.

HESITANT FUZZY SET THEORY APPLIED TO FILTERS IN MTL-ALGEBRAS

Abstract. The notions of a (Boolean, prime, ultra, good) hesitantfuzzy lter and a hesitant fuzzy MV - lter of an MTL-algebras areintroduced, and their relations are investigated. Characterizationsof a (Boolean, ultra) hesitant fuzzy lter are discussed. Conditionsfor a hesitant fuzzy set to be a hesitant fuzzy lter, and for a hesitantfuzzy lter to be a Boolean hesitant fuzzy lter are provided. 1. IntroductionMTL-algebras are the algebras of monoidal t-norm-based logic (MTL),as de ned by Esteva and Godo [2], and have BL-algebras as a subclass.The lter theory for MTL-algebras, which is also studied in several alge-braic structures, plays an important role in studying these algebras andthe completeness of the corresponding non-classical logics. From a log-ical point of view, various lters correspond to various sets of provableformulas. Borzooei et al. [1] studied several types of lters in MTL-algebras: EIMTL- lters, associative llters, IMTL- llters and strong llters. They provided several characterizations for these new types of lters and established the relations between them and the previouslyde ned implicative, positive implicative and fantastic lters in MTL-algebras as well as in BL-algebras.The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy setsand fuzzy multisets etc. are a generalization of fuzzy sets. As another

RANK AND PERIMETER PRESERVERS OF BOOLEAN RANK-1 MATRICES

For a Boolean rank-1 matrix A = ab t , we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the Boolean linear operators that preserve rank and perimeter of Boolean rank-1 matrices.

Matrices determined by a linear recurrence relation among entries

Abstract A matrix A =[ a ij ] is called a 7-matrix if its entries satisfy the recurrence relation αa i −1, j −1 + βa i −1, j = a ij where α , β are fixed numbers. A 7-matrix is completely determined by its first row and first column. In this paper we determine the structure of 7-matrices and investigate the sequences represented by columns of infinite 7-matrices.

Spanning column ranks and their preservers of nonnegative matrices

Abstract This paper concerns a certain column rank of matrices over the nonnegative reals; we call it the spanning column rank. We have a characterization of spanning column rank 1 matrices. We also investigate the linear operators which preserve the spanning column ranks of matrices over the nonnegative part of a certain unique factorization domain in the reals.

Linear operators that preserve maximal column rank of boolean matrices

This paper concerns two notions of column rank of matrices over semirings; column rank and maximal column rank. These two notions are the same over fields but differ for matrices over certain semirings. We determine how much the maximal column rank is different from the column ran for all m×n matrices over many semirings. We also characterize the linear operators which preserve the maximal column rank of Boolean matrices.

IDEMPOTENT MATRIX PRESERVERS OVER BOOLEAN ALGEBRAS

We consider the set of n × n idempotent matrices and we characterize the linear operators that preserve idempotent matrices over Boolean algebras. We also obtain characterizations of linear operators that preserve idempotent matrices over a chain semiring, the nonnegative integers and the nonnegative reals.

Zero-term rank preservers ∗

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.