Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If is generated as by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ arbitrary vector space $F$, the condition that $\Phi(x^2,x)=0 $ all $x\in A$ implies $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ $x,y,z\in A$. This applicable to question of whether zero Lie product determined and also used in pro...

Let A be an algebra over a commutative unital ring C. We say that A is zero triple product determined if for every C-module X and every trilinear map {·, ·, ·}, the following holds: if {x, y, z} 0 whenever xyz 0, then there exists a C-linear operator T : A3 −→ X such that {x, y, z} T xyz for all x, y, z ∈ A. If the ordinary triple product in the aforementioned definition is replaced by Jordan t...

Consider a Markovian Petri net with race policy. The marking process has a “product form” stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Zero Theorem of Feinberg, developed for chemical reaction networks, provides a structural and simple sufficie...

We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable 0-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of 0-automatic q...