Abstract.Suppose a Dedekind cr-complete partially ordered linear algebra (dsc-pola) satisfies a certain multiplication decomposition property (see definition below), then we show that this partially ordered linear algebra actually has the same structure of a special class of real matrix algebras, consisting of elements that can be decomposed as diagonal part plus nilpotent part w, such that w2=0.A dsc-pola, denoted by A (or B) is a real linear associative algebra which satisfies the following two conditions: (1) It is partially ordered so that it is a directed partially ordered linear space and O^xy whenever x, y e A, O^x, O^y. (1) It is Dedekind a-complete, i.e., if x" e A, 0^-• •■^x2^x1, then infix,,} exists. A dsc-pola A has the Archimedean property: If x, y e A and nx^y for every positive integer «, then x^O. In this paper we will assume A has a multiplicative identity 1^0. Let I={y:y^.l, and y1^0}czA.Define Ax = \JveI {x:-y^x^y}. Then it was shown by R.DeMarr that the multiplication of the elements in Ax is commutative, and Ax behaves much like an algebra of real-valued functions; moreover, Ax is a lattice and has no nonzero nilpotent. For the details of the proofs and examples of Ax we refer to [2]. (Note in [2], instead of the term dsc-pola, we use polac; actually they have the same meaning.) We will call Ax the functional or diagonal part of A. Let A be a dsc-pola which has the following multiplication decomposition property (abbreviated as MD):MD property: If yx, y2e A, Orgj,, 0^y2, 0
