Abstract
In Chapter X of his book on The structure of appearance, Goodman [1] studies the concept of linear order in finite linear arrays in terms of a two-place symmetrical predicate M, M(x, y) being read as x matches y. Although arising in the development of a nominalistic philosophical system, many problems relating to the properties of M are essentially of a mathematical and not of a philosophical nature. Some were considered by Fine in [2]. In the terminology of [1] and [2], the main result of this paper is that there is an effective method for embedding a weakly mapped array (consistently) in a uniform array. This carries with it the result that it is possible to obtain effectively a minimal uniform extension of the original array, that is, a uniform extension with minimum span for such an extension and with the minimum number of elements possible for a uniform extension with this span.
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