Hello pricey customer to our community We will proffer you an answer to this query reference request – A Hadamard product of binary (or ternary) matroids ,and the respond will breathe typical via documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

## reference request – A Hadamard product of binary (or ternary) matroids

I might love to know if anybody has studied the next “Hadamard product” of binary (or ternary) matroids. (There is a understanding of Hadamard product of matroids studied e.g. right here however I cerebrate that one is totally different.)

Let $M,N$ breathe unostentatious binary matroids of rank $r$ and $s$, respectively, over the identical floor clique $E$ of measurement $n$. For binary representations $(x_1,dots, x_n)$ and $(y_1,dots, y_n)$ of $M$ and $N$, respectively, outline the *Hadamard product* of $M circ N$ to breathe the binary matroid represented by $(x_1 otimes y_1, dots, x_n otimes y_n)$. One can simply display that this can be a well-defined matroid product, utilizing the very fact all representations of binary matroids are projectively equal [Proposition 6.6.5, *Matroid Theory*, Oxley].

After a miniature labor, one can emanate the linearly impartial units in $M circ N$. Suppose WLOG that $(x_1,dots, x_r)$ figure a foundation for $M$. For every $i in {1,dots, r}$, let

$$textual content{Supp}(i)={a in {1,dots, n} | x_a(i) neq 0},$$

the place $x_a(i)in mathbb{F}_2$ is the $i$-th coordinate of $x_a$ within the foundation $(x_1,dots, x_r)$. Then $S subseteq [n]$ is linearly impartial in $M circ N$ if and provided that for all $T subseteq [n]$ of measurement $1 leq |{T}| leq n-1$, there exists $i in {1,dots, r}$ such that $sum_{a in T} x_a(i) y_a neq 0$. This inequality is equal (over $mathbb{F}_2$), to maxim that the clique

$$T cap textual content{Supp}(i)$$ is just not Eulerian in $N$, i.e. it can’t breathe partitioned into circuits in $N$.

Along with a unostentatious request for details about this matroid product, I might breathe very fascinated by any suggestions on the next surmise, which is the $mathbb{F}_2$-version of a surmise I’ve been occupied with for a while (preprint right here).

**Conjecture:** Let $M_1,dots, M_m$ breathe unostentatious binary matroids of rank $r_1,dots, r_m$, respectively over the identical floor clique $E$ of measurement $n$. If $n leq sum_{j=1}^m (r_j-1)+1$, then $M_1 circ dots circ M_m$ is disconnected as a matroid.

I’ve confirmed this surmise when $m=2$; or $m=3$ and $r_3=2$; or $m$ is capricious, $r_1geq 1$ is capricious, and $r_2=dots=r_m=2$.

we’ll proffer you the answer to reference request – A Hadamard product of binary (or ternary) matroids query through our community which brings all of the solutions from a number of dependable sources.

## Add comment