### Fuzzy ideals in BE-algebras.

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The zero-term rank of a matrix is the minimum number of lines (row or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve the zero-term rank of the m × n integer matrices. That is, a linear operator T preserves the zero-term rank if and only if it has the form T(A)=P(A ∘ B)Q, where P, Q are permutation matrices and A ∘ B is the Schur product with B whose entries are all nonzero integers.

This paper concerns two notions of rank of fuzzy matrices: maximal column rank and column rank. We investigate the difference of them. We also characterize the linear operators which preserve the maximal column rank of fuzzy matrices. That is, a linear operator T preserves maximal column rank if and only if it has the form T(X) = UXV with some invertible fuzzy matrices U and V.

For a rank-1 matrix $A=a\otimes {b}^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T\left(A\right)=U\otimes {A}^{t}\otimes V$ with some monomial matrices U and V.

The notions of superior subalgebras and (commutative) superior ideals are introduced, and their relations and related properties are investigated. Conditions for a superior ideal to be commutative are provided.

The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb{M}}_{m,n}$. We call a matrix $A\in {\mathbb{M}}_{m,n}$ regular if there is a matrix $G\in {\mathbb{M}}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb{M}}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min\{m,n\}\le 2$, then all operators on ${\mathbb{M}}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, then an operator $T$ on ${\mathbb{M}}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T\left(X\right)=UXV$ for all $X\in {\mathbb{M}}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {\mathbb{M}}_{n}$.

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.

The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

Based on the hesitant fuzzy set theory which is introduced by Torra in the paper [12], the notions of Inf-hesitant fuzzy subalgebras, Inf-hesitant fuzzy ideals and Inf-hesitant fuzzy p-ideals in BCK/BCI-algebras are introduced, and their relations and properties are investigated. Characterizations of an Inf-hesitant fuzzy subalgebras, an Inf-hesitant fuzzy ideals and an Inf-hesitant fuzzy p-ideal are considered. Using the notion of BCK-parts, an Inf-hesitant fuzzy ideal is constructed. Conditions...

Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the $m\times n$ real matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.

For a rank-$1$ matrix $A={\mathbf{a}\mathbf{b}}^{t}$, we define the perimeter of $A$ as the number of nonzero entries in both $\mathbf{a}$ and $\mathbf{b}$. We characterize the linear operators which preserve the rank and perimeter of rank-$1$ matrices over semifields. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices over semifields if and only if it has the form $T\left(A\right)=UAV$, or $T\left(A\right)=U{A}^{t}V$ with some invertible matrices U and V.

Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of...

Let ${\mathbb{Z}}_{+}$ be the semiring of all nonnegative integers and $A$ an $m\times n$ matrix over ${\mathbb{Z}}_{+}$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m\times k$ matrix times a $k\times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2\times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the rank of $A$...

We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices if and only if it has the form $T\left(A\right)=P(A\circ B)Q$, or $T\left(A\right)=P({A}^{t}\circ B)Q$ with appropriate permutation matrices $P$ and $Q$ and positive integer matrix $B$, where $\circ $ denotes Hadamard product.

The notion of (implicative) int-soft ideal in a pseudo MV -algebra is introduced, and related properties are investigated. Conditions for a soft set to be an int-soft ideal are provided. Characterizations of (implicative) int-soft ideal are considered. The extension property for implicative int-soft ideal is established.

Given i, j, k ∈ {1,2,3,4}, the notion of (i, j, k)-length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

The notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an (anti) permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.

The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.

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