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Let $A, B$ be the algebras with two elements $0,1$ under addition mod $2$ and unary operation $x'=x$ in $A$ and $x'=1-x$ in $B$. Then $A \times B \cong B \times B$, though $A$ and $B$ are not isomorphic. Construct a 12-element commutative semigroup which does not have the unique factorization property. (exercise exercises on p. 141 170 of Birkhoff, Lattice Theory (3rd edition)) |
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Let $A, B$ be the algebras with two elements $0,1$ under addition mod $2$ and unary operation $x'=x$ in $A$ and $x'=1-x$ in $B$. Then $A \times B \cong B \times B$, though $A$ and $B$ are not isomorphic. (exercise on p. 141 of Birkhoff, Lattice Theory (3rd edition)) |
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