External Monoids
Since monoids are defined in terms of binary operations, we can define an external monoid in terms of external binary operations. For the sake of simplicity, unless otherwise specified, a left external binary operation is implied. Using the term external, we can make the generalizations:
- An external magma over R is a set S with an external binary operation. This satisfies for all (external closure).
- An external semigroup over is an external magma that satisfies for all (externally associative).
- An external monoid over is an external semigroup in which there exists such that for all (has external identity element).
Read more about this topic: External (mathematics)
Famous quotes containing the word external:
“All in all, the creative act is not performed by the artist alone; the spectator brings the work in contact with the external world by deciphering and interpreting its inner qualifications and thus adds his contribution to the creative act. This becomes even more obvious when posterity gives its final verdict and sometimes rehabilitates forgotten artists.”
—Marcel Duchamp (1887–1968)