The theory of the fields is a branch of mathematics whose principal field of application is in theoretical data processing. This part of the set theory ordered was introduced by Dana Scott during the Sixties, in order to provide the theoretical framework necessary to the definition of a dénotationnelle Sémantique of the Lambda-calculation.

The fields are partially ordered units. In the dénotationnelle semantics of lambda-calculation, the elements of the fields represent the lambda-terms, and the smallest element (when one provides the field with it) represents the result of a not finishing calculation, it is the element known as “indefinite”, noted $\backslash \; perp$ (to pronounce “bottom”). The order of the field defines, in the idea, a concept of quantity of information: an element of the field contains at least the contained information in the elements which are lower to him.

The idea is then to bring back itself to particular fields where any monotonous function (increasing) has a smaller fixed point. In general, one uses complete partial orders (partial order, or CPO supplements), i.e. fields which have a smaller element and where all chain (left strictly ordered) has an upper limit.

Thus, it becomes easy to associate a semantics with the controller of point fixes Y , by representing it by a total function which with a function associates one of its fixed points if there exists, and $\backslash \; perp$ if not. By là-même, to give a direction to a defined function “recursively” (i.e. in fact, as a fixed point of a functional calculus G ) becomes possible:

• if F is the function which with 0 associates 1 and with N > 0 associates N * F (N - 1) ,
• one can also define F as this: F = Y (G) (not fixes G ) where G is the function which takes a function φ in entry and returns the function which with 0 associates 1 and with N > 0 associates N * φ (N - 1) (and with $\backslash \; perp$ associates $\backslash \; perp$, by definition of $\backslash \; perp$). It is noted that G is monotonous on the field of the functions of ℕ_{$\backslash \; perp$} in ℕ_{$\backslash \; perp$}, and, for this reason, a point admits fixes (the factorial function!)
• then, one has a means of calculating F : by reiterating G on the function f₀ = $\backslash \; perp$ , i.e. the function which has entire naturalness and with $\backslash \; perp$ associates $\backslash \; perp$. F is the limit of the continuation thus obtained (and the smallest fixed point of G ).

The theory of the fields also makes it possible to give a direction to the equations of field of the $A\; type\; =\; has\; \backslash \; rightarrow\; A$ ( has is the whole of the functions of has in has !). In usual mathematics, this is absurd, unless giving a direction particular to this arrow. For example $\backslash \; reals\; =\; \backslash \; reals\; \backslash \; rightarrow\; \backslash \; reals$ appears impossible, would be this only for reasons of cardinality (In the theory of the cardinals $\backslash \; reals$ is a Infini strictly smaller than $\backslash \; reals\; \backslash \; rightarrow\; \backslash \; reals$); however, if this arrow represents only the continuous applications of $\backslash \; reals$ in $\backslash \; reals$, one keeps the same cardinal well as $\backslash \; reals$ (indeed, a continuous application of $\backslash \; reals$ in $\backslash \; reals$ can be defined by its restriction on the countable unit Q , therefore this unit has the cardinal of $\backslash \; reals\; ^\; \backslash \; omega$, therefore of $\backslash \; reals$).

In theory of the fields, the concept of continuity on a unit has will have its equivalent: continuity according to Scott on a field has . A function is Scott-continuous if it is monotonous on has and that for all left filter (left where any pair of elements has one raising) B has admitting an upper limit, there is $sup\; (F\; (B))\; =f\; (sup\; (B))$. This definition will be often simplified for the case where has is a CPO: the function is continuous if it is monotonous, and so for any chain B, there is $sup\; (F\; (B))\; =f\; (sup\; (B))$.