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

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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 $r_2=dots=r_m=2$.

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