Introduction. The von Neumann-Murray dimension theory of operator algebras [6] has been given an abstract setting by several authors, notably by Loomis [3] and Maeda [4] in the domain of lattice theory, and by Kaplansky in his theories of ^4W*-algebras [1] and Baer*-rings [2]. The equivalence classes of a dimension lattice possess many of the properties of a generalised cardinal algebra (GCA) [9] ; the purpose of this paper is to investigate this similarity. After a preliminary result on the axiomatization of GCA's we are able to show that the equivalence classes form a GCA. This is followed by an investigation of the structure of the resulting GCA's, leading to the representation of certain of them as algebras of continuous functions. Unfortunately, some of the dimension theory arguments needed here require completeness, whereas cardinal algebras have only cr-completeness. To overcome this difficulty we make the natural assumption, namely, the countable chain condition, with satisfactory results. We recall now some pertinent notions and results. A cardinal algebra (CA)is an algebraic system consisting of a set A, a binary operation + , and an operation Zof countable rank satisfying conditions 1.1 I—1.1 VII of [9]. The first five of these assert that A is closed under the operations, that Z is a generalization of + , that there is a zero element, and that the operations are unrestrictedly commutative and associative. VI Refinement Postulate. If a + b = Zef, then there are elements a0, ay , ■■■ and b0 , bx , ••• such that a = Hai, b = Zbi, and ai + bi = ci for all i. VII Remainder Postulate. Ifa„ = an + X + bn for all n, then there is an element c such that a„ = c + Hibn + ifor all n. (We make the agreement that, unless explicitly stated to the contrary, any index which appears will range over the non-negative integers.) In a generalized cardinal algebra (GCA) the operations need not always be defined, but conditions I-VII, when provided with suitable existential assumptions, are satisfied. Although we shall usually work with CA's , GCA's are perhaps more suitable for our purposes. This is because any w*-algebra of operators gives rise to a