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. identity Let be an algebraic variety, and let be an instance of . Suppose that instances of have a binary operation . Then an identity of with respect to is an element corresponding to a 0-ary operation of so that the rule holds, where is a variable. associativity Let be an algebraic variety whose instances have a binary operation . This binary relation is called associative iff the rule holds for . inverse Let be an algebraic variety whose instances have a binary operation and an identity . An inverse operation is a unary operation on so that the rule holds. commutativity Let be an algebraic variety with a binary relation . This binary relation is called commutative if and only if the rule holds. higher associativity Let be an algebraic variety with a binary relation . Suppose is associative. Then let be an instance of , and let . Let be a word of the first Dyck language .