Vol. 64, No. 2, 1976

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Plus and times

Richard Fairbanks Arnold and A. P. Morse

Vol. 64 (1976), No. 2, 297–332

Postulated here are very general but nevertheless concrete notions of plus and times. The generality achieved lessens the need for abstract linear spaces. The addition and multiplication are universally associative and commutative and multiplication rather widely distributes over addition.

The addable entities are of three general types: numbers, functions, and classes. The numbers are the finite complex numbers along with certain infinities. The addable functions are addable valued nonvacuous functions. The addable classes are free of numbers and ordered pairs; they are structured nonvacuous classes of addable entities. Many addable classes arise, as in §5, as congruence classes from an equivalence relation. In particular, the usual complex Lebesgue classes are among the addable classes.

Mathematical Subject Classification
Primary: 08A25, 08A25
Received: 10 April 1975
Published: 1 June 1976
Richard Fairbanks Arnold
A. P. Morse