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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.
B. Jonsson

Construct a 12-element commutative semigroup which does not have the unique factorization property.
R. McKinsey

(exercise exercises on p. 141 170 of Birkhoff, Lattice Theory (3rd edition))

1

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.
McKinsey

(exercise on p. 141 of Birkhoff, Lattice Theory (3rd edition))