Universal Algebra/Binary relations

Many algebraic varieties that are studied in mathematics have a very important binary relation. In this chapter, we study this case.

Definition (identity):

Let ${\mathcal {A}}$ be an algebraic variety, and let $A$ be an instance of ${\mathcal {A}}$. Suppose that instances of ${\mathcal {A}}$ have a binary operation $*$. Then an identity of $A$ with respect to $*$ is an element $e\in A$ corresponding to a 0-ary operation of ${\mathcal {A}}$ so that the rule $x*e=x$ holds, where $x$ is a variable.

Definition (associativity):

Let ${\mathcal {A}}$ be an algebraic variety whose instances have a binary operation $*$. This binary relation is called associative iff the rule $x*(y*z)=(x*y)*z$ holds for ${\mathcal {A}}$.

Definition (inverse):

Let ${\mathcal {A}}$ be an algebraic variety whose instances have a binary operation $*$ and an identity $e$. An inverse operation is a unary operation $~^{-1}$ on ${\mathcal {A}}$ so that the rule $x*x^{-1}=e$ holds.

Definition (commutativity):

Let ${\mathcal {A}}$ be an algebraic variety with a binary relation $*$. This binary relation is called commutative if and only if the rule $x*y=y*x$ holds.

Proposition (higher associativity):

Let ${\mathcal {A}}$ be an algebraic variety with a binary relation $*$. Suppose $*$ is associative. Then let $A$ be an instance of ${\mathcal {A}}$, and let $a_{1},\ldots ,a_{n}\in A$. Let $w\in D_{1}$ be a word of the first Dyck language $D_{1}$.