We consider *finite* algebras for a given signature, in the sense of universal algebra (for example, they might be groups, rings, or lattices). An algebra $A$ is *irreducible* when $A \cong B \times C$ implies that $B$ or $C$ is the one-point algebra.


Is it the case that a $\Sigma$-algebra can be expressed as a cartesian product of irreducible algebras in an essentially unique way, i.e., unique up to permutation of factors? I suspect that this is either a theorem with somebody's name attached to it, or there is a counterexample in groups.



Is the decomposition of an algebra into irreducible components essentially unique?
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 


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


