Introduction. Representation theory has contributed much to the study of linear associative algebras. The central problem of representation theory per se is the determination for each algebra of all its indecomposable representations. This turns out to be a much deeper problem than the classification of algebras, in the sense that there are algebras for which any "internal question" can be answered but for which the number and nature of representations is almost completely unknown, or if known

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... s much more complicated than the internal theory. This can be illustrated by the example of a commutative algebra of order three for which the representation theory can be shown to be essentially the same as the problem of classifying pairs of rectangular matrices under equivalence. (This algebra has indecomposable representations of every integral degree.) Detailed study (as yet unpublished) of the representations of certain classes of algebras has led me to consider the possibility of searching for connections between representation theory and lattice theory. The present note is devoted to setting up the machinery for certain phases of such an investigation. Notation and definitions. Let t be a sfield and Vs. right f-space of dimension n. If vi t . . . , v n are a basis for V then any vector v in V can be written in the form v = Vid\ + . . . + v n a n , where a^ . . . , a n are uniquely determined scalars (i.e. elements of f) called the coordinates of v relative to the given basis for V. This can be written in the matrix form as v = lri|r|p<|| where \\vj\\ denotes the 1 by n (row) matrix made up of the basis vectors and ||a*|| denotes the n by 1 (column) matrix made up of the coordinates. (In describing any matrix we shall use the subscript "i" for row index and "j" for column index.) Then for any vector v and scalar a, va is the vector with coordinate matrix \\b\\ -\\aia]\. We denote by X the set of all linear transformations a (i.e., f-endomorphism) of V into itself. We write the linear transformations as left operators, and then the commutativity of linear transformations with the scalar multiplications take the form (av)a = a (va). To express the linear transformations in matrix form we use the formula «» = a(|hlHM|) = («|M|)||a,|| = iKlHkll = (lhl|r.)|M| = IMI(r.||a f ||). Here T a is, of course the matrix whose jth column is the coordinate matrix of