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)?



Posets (partially ordered sets) in equational logic
No. The category of models of an equational theory (i.e. a variety in the sense of universal algebra) is always a [regular category](https://ncatlab.org/nlab/show/regular+category), but the category of posets is not regular.


