Let $F$ be a subfield of the field $ℝ$ of real numbers. Equipped with the binary arithmetic mean operation, each convex subset $C$ of $F^n$ becomes a commutative binary mode, also called idempotent commutative medial (or entropic) groupoid. Let $C$ and $C^{\text{'}}$ be convex subsets of $F^n$. Assume that they are of the same dimension and at least one of them is bounded, or $F$ is the field of all rational numbers. We prove that the corresponding idempotent commutative medial groupoids are isomorphic iff the affine space $F^n$...

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $𝒜$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $𝒟\left(A\right)$ of all deductive systems on $𝒜$. Moreover, relative annihilators of $C\in 𝒟\left(A\right)$ with respect to $B\in 𝒟\left(A\right)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $𝒟\left(A\right)$.

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.