The idea of an order function originated with the second author [\textit{R. Pellikaan}, J. Stat. Plann. Inference 94, No.~2, 287-301 (2001; Zbl 0981.94055)] as a way to consider one-point Goppa codes using ring theory, without using algebraic geometry. An order function generalizes the order of pole of a rational function along a prime divisor. The concepts of order function and order domain were then generalized by \textit{R. Matsumoto} and \textit{S. Miura} [``On construction and generalization of algebraic geometry codes'', in: Proc. Algebr. Geom., Number Theory, Coding Theory and Cryptography, Univ. Tokyo 2000, 3-15 (2000)], and by \textit{M. E. O'Sullivan} [Finite Fields Appl. 7, No. 2, 293-317 (2001; Zbl 1027.94032)]. Here, the authors further generalize these concepts, with an order structure consisting of an algebra \(R\) (order domain) over a field and a map (order function), satisfying certain properties, from \(R\) to a well-ordered set, which then gets a semigroup structure from the order function. If this semigroup is finitely generated, then the authors use Hilbert functions to show that the rank of this semigroup is equal to the dimension of the ring \(R\). Following \textit{M. E. O'Sullivan} [loc. cit.], the authors extend the theory of Gröbner bases to order domains. They also study the behavior of order domains under formation of factor rings, extension of scalars, and tensor product.