# Direct product is cancellative for finite algebras in any variety with zero

## Contents

## Statement

Suppose is a variety of algebras that is a variety with zero. Suppose are *finite* algebras in . Further, suppose that the algebras and are isomorphic as algebras in . Then, is isomorphic to .

## Related facts

- Direct product is cancellative for finite groups: Essentially the same proof, tailored to groups (the key specific detail for groups is that the congruences can be described using the normal subgroups that function as their kernels).

## Facts used

- Homomorphism set to direct product is Cartesian product of homomorphism sets: If are algebras, then there is a natural bijection:
- .
- The bijection is defined as: .

- Homomorphism set is disjoint union of injective homomorphism sets: For algebras and , let denotes the set of homomorphisms from to , and denote the set of injective homomorphisms from to . Then we have:

.

Here varies over the set of all possible congruences on the algebra .

## Proof

**Given**: Finite algebras such that .

**To prove**: .

**Proof**: Let be an arbitrary finite algebra in . Note that the trivial homomorphism refers to the map that sends every element to the zero element.

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | as an equality of finite numbers. | Fact (1) | are finite | [SHOW MORE] | |

2 | as an equality of finite numbers. | Fact (1) | are finite | [SHOW MORE] | |

3 | as an equality of finite numbers. | The number of homomorphisms to an algebra depends only on its isomorphism type. | |||

4 |
as an equality of finite numbers. | Steps (1),(2),(3) | [SHOW MORE] | ||

5 | For any finite algebra , the number of injective homomorphisms from to equals the number of injective homomorphisms from to . We show this by induction on the order of . In other words, | Fact (2) | Step (4) | [SHOW MORE] | |

6 | is isomorphic to a subalgebra of and is isomorphic to a subalgebra of | are finite | Step (5) | [SHOW MORE] | |

7 | is isomorphic to | are finite | Step (6) | [SHOW MORE] |