Source : Free On-Line Dictionary of Computing

domain theory
     
         A branch of mathematics introduced by Dana Scott in
        1970 as a mathematical theory of programming languages, and
        for nearly a quarter of a century developed almost exclusively
        in connection with {denotational semantics} in computer
        science.
     
        In {denotational semantics} of programming languages, the
        meaning of a program is taken to be an element of a domain.  A
        domain is a mathematical structure consisting of a set of
        values (or "points") and an ordering relation, <= on those
        values.  Domain theory is the study of such structures.
     
        ("<=" is written in {LaTeX} as {\subseteq})
     
        Different domains correspond to the different types of object
        with which a program deals.  In a language containing
        functions, we might have a domain X -> Y which is the set of
        functions from domain X to domain Y with the ordering f <= g
        iff for all x in X, f x <= g x.  In the {pure lambda-calculus}
        all objects are functions or {application}s of functions to
        other functions.  To represent the meaning of such programs,
        we must solve the {recursive} equation over domains,
     
        	D = D -> D
     
        which states that domain D is ({isomorphic} to) some {function
        space} from D to itself.  I.e. it is a {fixed point} D = F(D)
        for some operator F that takes a domain D to D -> D.  The
        equivalent equation has no non-trivial solution in {set
        theory}.
     
        There are many definitions of domains, with different
        properties and suitable for different purposes.  One commonly
        used definition is that of Scott domains, often simply called
        domains, which are {omega-algebraic}, {consistently complete}
        {CPO}s.
     
        There are domain-theoretic computational models in other
        branches of mathematics including {dynamical systems},
        {fractals}, {measure theory}, {integration theory},
        {probability theory}, and {stochastic processes}.
     
        See also {abstract interpretation}, {bottom}, {pointed
        domain}.
     
        (1999-12-09)