Coalgebras for an endofunctor provide a category theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired by and resembles the standard breadth-first search procedure to compute the reachable part of a graph. We also study coalgebras in Kleisli categories: for a functor extending a functor on the base category, we show that the reachable part of a given pointed coalgebra...

Given a groupoid $\langle G,☆\rangle$, and $k\ge 3$, we say that $G$ is antiassociative if an only if for all $x_1,x_2,x_3\in G$, $(x_1☆x_2)☆x_3$ and $x_1☆(x_2☆x_3)$ are never equal. Generalizing this, $\langle G,☆\rangle$ is $k$-antiassociative if and only if for all $x_1,x_2,...,x_k\in G$, any two distinct expressions made by putting parentheses in $x_1☆x_2☆x_3☆\cdots ☆x_k$ are never equal. We prove that for every $k\ge 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.