Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

We will investigate so-called commuting operators and their relationship to bisymmetry and domination. In the case of bisymmetric aggregation operators we will show a sufficient condition ensuring that two operators commute, while for bisymmetric aggregation operators with neutral element we will even give a full characterization of commuting n-ary operators by means of unary commuting operators.

We show that for all k ≥ 1, there exists an integer N(k) such that for all n ≥ N(k) the k-th order jet scheme over the commuting n × n matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all k ≥ 1, the k-th order jet scheme over the commuting 2× 2 matrices is irreducible: we show that the same holds for n = 3.

‎in a recent paper c‎. ‎miguel proved that the diameter of the commuting graph of the matrix ring $mathrm{m}_n(mathbb{r})$ is equal to $4$ if either $n=3$ or $ngeq5$‎. ‎but the case $n=4$ remained open‎, ‎since the diameter could be $4$ or $5$‎. ‎in this work we close the problem showing that also in this case the diameter is $4$.

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

The result from the title is shown. Let L(H) denote the algebra of all bounded linear operators on a complex Hilbert space H. If M⊂ L(H), then we denote by M′ the commutant of M, M′ = {S ∈ L(H) : T S = S T for every T ∈ M}. The second commutant is denoted by M′′ = (M′)′. Denote further by W(M) the smallest weakly closed subalgebra of L(H) containing M and by AlgLatM the algebra of all operators...