Variety of lattices

Example

Take the signature with two binary operations $\vee$ and $\wedge$. The variety of lattices is the class of algebras $(L, \vee, \wedge)$ with the identities $$x \vee y = y \vee x, \qquad (x \vee y) \vee z = x \vee (y \vee z), \qquad x \vee (x \wedge y) = x,$$ together with the same three identities with $\vee$ and $\wedge$ exchanged. These algebras are exactly the lattices. The order is recovered by $x \le y$ if and only if $x \vee y = y$. In this order, $x \vee y$ and $x \wedge y$ are the join and the meet. The class is a variety, since it is defined by identities.