### A counting theorem in the semigroup of circulant Boolean matrices

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|{q}_{0}|=...=|{q}_{N-1}|=1$ the quantity $\Phi ={\sum}_{i+k=j+l}\frac{{q}_{i}{q}_{k}}{{q}_{j}{q}_{l}}$ satisfies $\Phi \ge {N}^{2}$, with equality if and only if $q=\left({q}_{i}\right)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi $, (2) the study of the critical points of $\Phi $, and (3) the computation of the moments of $\Phi $. We explore here these questions,...

Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n2/4]. In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition,we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms.

Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that ${A}^{k}$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).

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 $X\in {\mathbb{M}}_{n}$.

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...

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 ${\mathbb{B}}_{k}$ be the general Boolean algebra and $T$ a linear operator on ${M}_{m,n}\left({\mathbb{B}}_{k}\right)$. If for any $A$ in ${M}_{m,n}\left({\mathbb{B}}_{k}\right)$ (${M}_{n}\left({\mathbb{B}}_{k}\right)$, respectively), $A$ is regular (invertible, respectively) if and only if $T\left(A\right)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over ${\mathbb{B}}_{k}$. Meanwhile, noting that a general Boolean algebra ${\mathbb{B}}_{k}$ is isomorphic...

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.

We characterize matrices whose powers coincide with their Hadamard powers.