noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $math...

Let $Q$ be an inverse semigroup. A subsemigroup $S$ of $Q$ is a left I-order in $Q$ and $Q$ is a semigroup of left I-quotients of $S$ if every element $qin Q$ can be written as $q=a^{-1}b$ for some $a,bin S$. If we insist on $a$ and $b$ being $er$-related in $Q$, then we say that $S$ is straight in $Q$. We characterize semigroups which are left I-quotients of left regular bands of right cancell...

The q-Dobinski formula may be interpreted as the average of powers of random variable X q with the q-Poisson distribution. Forty years ago Rota G. C. [1] proved the exponential generating function for Bell numbers B n to be of the form ∞ n=0 x n n! (B n) = exp(e x − 1) (1) using the linear functional L such that L(X n) = 1, n ≥ 0 (2) Then Bell numbers (see: formula (4) in [1]) are defined by L(...