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Representations and completions for ordered algebraic structures

The primary concerns of this thesis are completions and representations for various classes of 
poset expansion, and a recurring theme will be that of axiomatizability. By a representation we 
mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically 
embedded into a field of sets. So, in general we are interested in order embedding 
posets into fields of sets in such a way that existing meets and joins are interpreted naturally as 
set theoretic intersections and unions respectively. 
Our contributions in this area are an investigation into the ostensibly second order property 
of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets 
and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among 
other things we show that unlike Boolean algebras, which have such a ‘complete’ representation 
if and only if they are atomic, the classes of bounded, distributive lattices and posets with 
complete representations have no first order axiomatizations (though they are pseudoelementary). 
We also show that the class of posets with representations preserving arbitrary joins is 
pseudoelementary but not elementary (a dual result also holds). 
We discuss various completions relating to the canonical extension, whose classical construction 
is related to the Stone representation. We claim some new results on the structure of 
classes of poset meet-completions which preserve particular sets of meets, in particular that they 
form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions 
using a two stage process involving meet- and join-completions. 
Linking our twin topics we discuss canonicity for the representation classes we deal with, 
and by building representations using a meet-completion construction as a base we show that 
the class of representable ordered domain algebras is finitely axiomatizable. Our method has 
the advantage of representing finite algebras over finite bases.