TITLE: Is a poset an algebra?
QUESTION [12 upvotes]: I encountered some nice material about so-called universal algebras.
E.g. a lattice is an algebra (in the sense of universal algebra), and can be presented as a tuple $\langle L,\wedge,\vee\rangle$ where $\wedge$ and $\vee$ are binary operations on $L$.
I am not (yet) familiar with that stuff, and one of the first questions that arose was:

Can a partial order be presented as an algebra?

I think not because I cannot really find operations on it with some (finite) arity. 
Am I correct in my thinking?

More generally: if I see the objects of some category as candidate for being an algebra of some type, then are there criteria that can be applied in order to check this?

REPLY [5 votes]: From the question:

More generally: if I see the objects of some category as candidate for being an algebra of some type, then are there criteria that can be applied in order to check this?


There is a theorem of algebra that says
Theorem. A bijective morphism is an isomorphism.
That is, if $\varphi: A\to B$ is a homomorphism that is 1-1 and onto, then $\varphi^{-1}$ is also a homomorphism. More categorically, the theorem says that, if $\mathcal C$ is any full subcategory of the category of all algebras of some signature, then the forgetful functor to Sets reflects isomorphisms.
This theorem states a simple necessary condition for a concrete category to be equivalent to a concrete category of algebras. To apply it to the category of posets, observe that there is an order-preserving bijection between the 2-element discrete poset (=totally unordered poset) and the 2-element chain. Its inverse is not order preserving. 
You can easily apply this criterion to many other concrete categories, like the categories of graphs, topological spaces, ETC.