##  [Subdirectly Irreducible Algebra](/subdirectly-irreducible-algebra) 

  ##  [Subdirectly Irreducible Algebra](https://puremath.quantumdictionary.io/subdirectly-irreducible-algebra-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra is subdirectly irreducible (SI) if it cannot be represented nontrivially as a subdirect product of other algebras; equivalently, it has a least nontrivial congruence (the monolith), i.e., among congruences distinct from the diagonal there is a unique minimal one.

 

 

 

 

 





 

 



 ##  [Subdirectly Irreducible Algebra](https://algebra.quantumdictionary.io/subdirectly-irreducible-algebra-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebra that cannot be represented as a nontrivial subdirect product of other algebras; equivalently (in varieties) it has a unique minimal nontrivial congruence (the monolith).