### A counterexample in the cohomology of monoids.

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Let $F$ be a subfield of the field $\mathbb{R}$ 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 $\mathcal{A}$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $\mathcal{D}\left(A\right)$ of all deductive systems on $\mathcal{A}$. Moreover, relative annihilators of $C\in \mathcal{D}\left(A\right)$ with respect to $B\in \mathcal{D}\left(A\right)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $\mathcal{D}\left(A\right)$.

Given a groupoid $\langle G,\u2606\rangle $, and $k\ge 3$, we say that $G$ is antiassociative if an only if for all ${x}_{1},{x}_{2},{x}_{3}\in G$, $({x}_{1}\u2606{x}_{2})\u2606{x}_{3}$ and ${x}_{1}\u2606({x}_{2}\u2606{x}_{3})$ are never equal. Generalizing this, $\langle G,\u2606\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 ${x}_{1}\u2606{x}_{2}\u2606{x}_{3}\u2606\cdots \u2606{x}_{k}$ 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.

In a groupoid, consider arbitrarily parenthesized expressions on the $k$ variables ${x}_{0},{x}_{1},\cdots {x}_{k-1}$ where each ${x}_{i}$ appears once and all variables appear in order of their indices. We call these expressions $k$-ary formal products, and denote the set containing all of them by ${F}^{\sigma}\left(k\right)$. If $u,v\in {F}^{\sigma}\left(k\right)$ are distinct, the statement that $u$ and $v$ are equal for all values of ${x}_{0},{x}_{1},\cdots {x}_{k-1}$ is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds...