A homomorphism is a mapping between two algebraic systems of the same type¹ (one source mentions only monoids² while another mentions groups, rings or vectors³) such that:

1. Results obtained from all operations possible with the first system are mapped to results obtained from all operations possible with the second system.
2. The identity element(s) (with associated operator) is mapped from the first system to the second.

(Also see structures.)

1 McGraw-Hill dictionary of Physics and Mathematics, 1978, p. 458.

2 Michael A. Arbib, A. J. Kfoury and Robert N. Moll, A Basis for Theoretical Computer Science (New York: Springer-Verlag, 1981), p. 47.

3 Webster's New Collegiate Dictionary, 1975, p. 548.

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