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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)?

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    $\begingroup$ The algebraic representation in the question works just the same for semilattices. One can generalize it further to arbitrary directed posets, which can be algebraically represented by directoids (groupoids satisfying the identities $xx=x$, $((xy)z)x=(xy)z$, $y(xy)=xy$; one can restrict attention to commutative ones if desired), however, this is not so nice, as several distinct directoids may correspond to the same poset. $\endgroup$
    – Emil Jeřábek
    Commented Apr 6, 2016 at 14:51
  • $\begingroup$ You can also look for classes of posets that can be so represented. This comprises, for instance, all posets $(A;\leq)$ such that there is a binary operation $\cdot$ on $A$ with $a\cdot b = a$ iff $a\leq b$. For instance, in this paper by Gerhard, it is shown that posemigroups are not residually small (lots of them). $\endgroup$
    – Pedro Sánchez Terraf
    Commented Apr 8, 2016 at 11:46

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No. The category of models of an equational theory (i.e. a variety in the sense of universal algebra) is always a regular category, but the category of posets is not regular.

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  • $\begingroup$ It's nice. It also resolves the cases of preorders and topological spaces. $\endgroup$
    – H Koba
    Commented Apr 8, 2016 at 2:53

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