By Kenneth E. Iverson

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**Example text**

Thus (111 X 11) C ~- '~ (111 (111 C +---~ (111 27 positional nUlnher ,\ysteI11S X 11) <=> C/ <=> C: + 1 _ X 11) <--:? i In i 11 (111, i 11 ) ) for i andj == 1, 2, == 1,2, , 111 ,11. J1) <=> C~: ll~ J The use of the matrices E and I will be illustrated briefly. The relation == 2 10 (u v) can be extended to logical matrices as follovvs: u ;, v U / V == (2 E) 10 ( u ~ V) ~ the trace of a square numerical matrix X may be expressed as t == The triangular matrices are employed in the succeeding section.

The extension of the maximum prefix operation to the rows of a logical nlatrix V is denoted by 'XI V and defined as the compatible logical Illatrix V, such that Vi == 'XI Vi. The corresponding maximuIn colunln prefix operation is denoted by 'XII V. Right justification of a n uIllerical Illatrix X is achieved by the rotation k t X, where k == +IU)/(X == 0), and top just{jication is achieved by the rotation (+ 11'XII(X == 0)) 11 X (see Sec. ) A vector whose components are all distinct will be called an ordered set.

Xi For example, (YE) 1 X represents a set of polynomials in ?! with coeffi cients Xl' X 2 , ••• , Xl" and Y l X represents a set of eval uations of the vector x in a set of bases yl, y2, ... , Y/l. 15 SET OPERATIONS In conventional treatments, such as Jacobson (1951) or Birkhoff and Mac Lane (1941), a set is defined as an unordered collection of distinct elenlents. A calculus of sets is then based on such elementary relations as set membership and on such elementary operations as set intersection and set union, none of which imply or depend on an ordering anlong mernbers of a set.