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A '''triadic relation''' is a relation on an ordered triple of nonempty sets.  Thus, <math>L\!</math> is a triadic relation on <math>(X, Y, Z)\!</math> if and only if <math>L \subseteq X \times Y \times Z.\!</math>  Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation <math>L \subseteq X \times Y \times Z</math> to refer to a logical predicate or a propositional function, of the type <math>X \times Y \times Z \to \mathbb{B},</math> or any one of the derived binary operations, of the three types <math>X \times Y \to \operatorname{Pow}(Z),</math> <math>X \times Z \to \operatorname{Pow}(Y),</math> and <math>Y \times Z \to \operatorname{Pow}(X).</math>

A '''triadic relation''' is a relation on an ordered triple of nonempty sets.  Thus, <math>L\!</math> is a triadic relation on <math>(X, Y, Z)\!</math> if and only if <math>L \subseteq X \times Y \times Z.\!</math>  Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation <math>L \subseteq X \times Y \times Z</math> to refer to a logical predicate or a propositional function, of the type <math>X \times Y \times Z \to \mathbb{B},</math> or any one of the derived binary operations, of the three types <math>X \times Y \to \operatorname{Pow}(Z),</math> <math>X \times Z \to \operatorname{Pow}(Y),</math> and <math>Y \times Z \to \operatorname{Pow}(X).</math>

A '''binary operation''' or a '''law of composition''' (LOC) on a nonempty set <math>X\!</math> is a triadic relation <math>* \subseteq X \times X \times X\!</math> that is also a function <math>* : X \times X \to X.\!</math>  The notation <math>{}^{\backprime\backprime} x * y {}^{\prime\prime}\!</math> is used to indicate the functional value <math>*(x, y) \in X,\!</math> which is also referred to as the '''product''' of <math>x\!</math> and <math>y\!</math> under <math>*.\!</math>

A '''binary operation''' or '''law of composition''' (LOC) on a nonempty set <math>X\!</math> is a triadic relation <math>* \subseteq X \times X \times X\!</math> that is also a function <math>* : X \times X \to X.\!</math>  The notation <math>{}^{\backprime\backprime} x * y {}^{\prime\prime}\!</math> is used to indicate the functional value <math>*(x, y) \in X,\!</math> which is also referred to as the '''product''' of <math>x\!</math> and <math>y\!</math> under <math>*.\!</math>

A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''associative''' if and only if <math>(x*y)*z = x*(y*z)\!</math> for every <math>x, y, z \in X.\!</math>

A binary operation or LOC * on X is "associative" if and only if (x*y)*z = x*(y*z) for every x, y, z C X.

A binary operation or LOC * on X is "commutative" if and only if x*y = y*x for every x, y C X.

A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''commutative''' if and only if <math>x*y = y*x\!</math> for every <math>x, y \in X.\!</math>

A "semigroup" consists of a nonempty set with an associative LOC on it.  On formal occasions, a semigroup is introduced by means a formula like "X = <X, *>", read "X is the ordered pair <X, *>".  This form specifies X as the nonempty set and * as the associative LOC.  By way of recalling the extra structure, this specification underscores the name of the set X to form the name of the semigroup X.  In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set.  In contexts where more than one semigroup is formed on the same set, one can use notations like Xi = <X, *i> to distinguish them.

A "semigroup" consists of a nonempty set with an associative LOC on it.  On formal occasions, a semigroup is introduced by means a formula like "X = <X, *>", read "X is the ordered pair <X, *>".  This form specifies X as the nonempty set and * as the associative LOC.  By way of recalling the extra structure, this specification underscores the name of the set X to form the name of the semigroup X.  In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set.  In contexts where more than one semigroup is formed on the same set, one can use notations like Xi = <X, *i> to distinguish them.