I know about equational logic, cf. https://en.wikipedia.org/wiki/Lattice_(order)#As_algebraic_structure, and understood that lattices are expressed equationally, i.e., in terms of equational logic (with function symbols $\wedge, \vee$ and by introducing the order $p \le q \; :\Leftrightarrow \; p = p \wedge q$).

The question is, can posets be expressed equationally (by some function symbols and an order determined by them)?