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Question1 Definition 1: Let A and B be sets

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Question1 Definition 1: Let A and B be sets. A relation ? from A to B is a subset of A×B. . Define a?={b?B?(a,b)??} for each a?A Definition 2: If S and T are semigroups, then ??S×T is a relational morphism from S to T if (?a?S) a??? (RM1) (a?)(b?)?(ab)?.(RM2) Further, we say ? is injective if (?a,b?S)a??b????a=b Definition 3: We say S divides T , written S?T ,if there exists a subsemigroup U of T and a morphism ? from U onto S. Show that: S divides T if and only if there exists an injective relational morphism from S to T. Question 2 A semigroup S is called an inflation of one of its subsemigroups T if S^2 ? T and there is a surjection ? : S ? T such that ?^2 = ? and (?(a))(?(b)) = ab for all a, b ? S. Show that S is an inflation of a semilattice if and only if xy = y^2x for all x, y ? S.