### Table of Contents

## Relation algebras

Abbreviation: **RA**

### Definition

A \emph{relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\circ,^{\smile},e\rangle$ such that

$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a Boolean algebra

$\langle A,\circ,e\rangle $ is a monoid

$\circ$ is \emph{join-preserving}: $(x\vee y)\circ z=(x\circ z)\vee (y\circ z)$

$^{\smile}$ is an \emph{involution}: $x^{\smile\smile}=x$, $(x\circ y)^{\smile}=y^{\smile}\circ x^{\smile}$

$^{\smile}$ is \emph{join-preserving}: $(x\vee y)^{\smile}=x^{\smile}\vee y^{\smile}$

$\circ$ is residuated: $x^{\smile}\circ(\neg (x\circ y))\le\neg y$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be relation algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\circ$, $^{\smile}$, $e$:

$h(x\circ y)=h(x)\circ h(y)$, $h(x^{\smile})=h(x)^{\smile}$, $h(e)=e$

### Examples

Example 1: $\langle \mathcal P(U^2), \cup, \emptyset, \cap, U^2, -, \circ, ^\smile, id_U \rangle$ the full relation algebra of binary relations on a set $U$.

Example 2: $\langle \mathcal P(G), \cup, \emptyset, \cap, G, -, \circ, ^\smile, \{e\} \rangle$ the group relation algebra of a group $\langle G, *, ^{-1}, e \rangle$, where $X\circ Y=\{x*y : x\in X, y\in Y\}$ and $X^\smile=\{x^{-1} : x\in X\}$.

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | undecidable |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | yes, $n=2$ |

Congruence regular | yes |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | yes |

Equationally def. pr. cong. | yes |

Discriminator variety | yes |

Amalgamation property | no |

Strong amalgamation property | no |

Epimorphisms are surjective | no |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &0

f(4)= &3

f(5)= &0

f(6)= &0

\end{array}$