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 present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.

We prove that an orthomodular lattice can be considered as a groupoid with a distinguished element satisfying simple identities.

A (finite) acyclic connected graph is called a tree. Let $W$ be a finite nonempty set, and let $H\left(W\right)$ be the set of all trees $T$ with the property that $W$ is the vertex set of $T$. We will find a one-to-one correspondence between $H\left(W\right)$ and the set of all binary operations on $W$ which satisfy a certain set of three axioms (stated in this note).

Characterizations of 'almost associative' binary operations generating a minimal clone are given for two interpretations of the term 'almost associative'. One of them uses the associative spectrum, the other one uses the index of nonassociativity to measure how far an operation is from being associative.

We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V\left(G\right)$ and $uv\in E\left(G\right)$ if and only if $u\ne v$, $u*v=v$ and $v*u=u$ for any $u$, $v\in V\left(G\right)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).

Term substitution induces an associative operation on the free objects of any equational variety. In the case of left distributivity, the construction can be extended to any monogenic structure.

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.

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...

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.