An ordered field can be described relationally $(S,<,+,*)$ or it can be described purely algebraically $(S,\wedge,\vee,+,*)$. In this way, there are four different operations associated with the ordered field.
- Min: the meet of the total order
- Max: the join of the total order
- Addition: the addition of the field
- Multiplication: the multiplication of the field
The axioms for the four algebraic operations are the field axioms for addition and multiplication, the lattice axioms for the minimum and maximum operations and the totality of the two operations which means that for any two elements $a$ and $b$ the minimum and maximum both produce one of the two elements. For non-total orders the lattice operations can produce values that are not contained in the total order. In addition to this, there is the additive monotonicity and positivity preservation which can easily be translated to the algebraic formulation. A result of this description is that all the operations are
commutative.
Commutative structures
What algebraic structures can we form if we put the limitation that all operations have to be commutative? This is purely interesting as a thought experiment and to gather information on commutative semigroups. We can then form a restricted
outline of algebraic structures. We can exclude vector spaces for now because scalar multiplication is typically not defined to be commutative.
Single binary operation:
These structures are all of the form $(S,*)$ where $*$ is some commutative semigroup. The most important particular cases are the commutative groups and commutative group-free (aperiodic) semigroups. That is to say, the most important property is rather then semigroup is a group or it has any groups embedded in it. So we already split up our ontology into two special cases. Semilattices are a special case of aperiodic semigroups because they are group-free.
Two commutative binary operations:
When we are given two commutative binary operations like $+$ and $*$ generally speaking they are going to be related by the distributive law, which connects them so that we don't need to view them separately. This leads to commutative semigroup pairs, commutative hemirings, commutative rings, integral domains, fields, etc. All the structures of commutative algebra are in this category. The other case is that the operations are related by the absorportion law which produces lattices, modular lattices, distributive lattices, etc.
Four commutative binary operations:
Finally we get at ordered fields which as we described at the start can actually be described entirely using commutative operations. Ordered fields and related structures are the most advanced structures I know of that can be formed entirely with commutative operations as they combine both lattice and field structures.
