Let $𝒳$ be a Banach space of dimension $n>1$ and $𝔄\subset ℬ\left(𝒳\right)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:𝔄\to 𝔄$ (not necessarily linear) satisfies $$d\left(\right[[U,V],W\left]\right)=\left[\right[d\left(U\right),V],W]+\left[\right[U,d\left(V\right)],W]+\left[\right[U,V],d(W\left)\right]$$ for all $U,V,W\in 𝔄$, then $d=\psi +\tau$, where $\psi$ is an additive derivation of $𝔄$ and $\tau :𝔄\to 𝔽I$ vanishes at second commutator $\left[\right[U,V],W]$ for all $U,V,W\in 𝔄$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in 𝔄$ and a linear mapping $\tau$ from $𝔄$ into $𝔽I$ satisfying $\tau \left(\right[[U,V],W\left]\right)=0$ for all $U,V,W\in 𝔄$, such that $d\left(U\right)=SU-US+\tau \left(U\right)$ for all $U\in 𝔄$.

Semirings are modifications of unitary rings where the additive reduct does not form a group in general, but only a monoid. We characterize multiplicatively idempotent semirings and Boolean rings as semirings satisfying particular identities. Further, we work with varieties of enriched semirings. We show that the variety of enriched multiplicatively idempotent semirings differs from the join of the variety of enriched unitary Boolean rings and the variety of enriched bounded distributive lattices....

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite...

In the cases $𝔸$ and $\widetilde{𝔸}$, we describe the seeds obtained by sequences of mutations from an initial seed. In the $\widetilde{𝔸}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.