Let M be a class of multiplicative sets (cf. Chapter 2.3), and let A be any multiplicative set. A homomorphism α of A into some set belonging to M is called a representation of A in the class M. The set α (A) is called the image of the representation. If M consists of one multiplicative set M, we say that α is a representation of A in M instead of saying “a representation of A in the class consisting of the one set M.” If α is an isomorphism, the representation is said to be faithful. A representation in a class of semigroups of transformations is called a representation by transformations (or by permutations).
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