In abstract algebra an action of a set on a set is a mapping of into , the set of functions of into itself. When there is no risk of confusion, the element , for and , is sometimes denoted , or .
Let be sets, and let and be actions of on and , respectively. An -morphism of into is a function for which , for all in .
Let be sets, a function of into , an action of on , and an action of on . A mapping is called a -morphism if for all in and in . If is the identity map of , then the terms "-morphism" and "-morphism" are synonymous.
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N. Bourbaki, Algebra: Ch. 1–3, Springer, 1989, ISBN 3-540-64243-9 .