### Table of Contents

## Integral relation algebras

Abbreviation: **IRA** (this may also abbreviate the variety generated by all integral relation algebras)

### Definition

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

\emph{integral}: $x\circ y=0\Longrightarrow x=0\mbox{ or }y=0$

### Definition

An \emph{integral relation algebra} is a relation algebra $\mathbf{A}=\langle A,\vee,0, \wedge,1,',\circ,^{\smile},e\rangle$ in which

\emph{the identity element $e$ is $0$ or an atom}: $e=x\vee y\Longrightarrow x=0\mbox{ or }y=0$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be integral relation algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\circ y)=h(x)\circ h(y)$, $h(x\vee y)=h(x)\vee h(y)$, $h(x')=h(x)'$, $h(x^\smile)=h(x)^\smile$ and $h(e)=e$.

### Examples

For any group $\mathbf G=\langle G,*,^{-1},e\rangle$, construct the integral relation algebra $\mathcal R(G)=\langle\mathcal P(G),\cup,\emptyset,\cap,G,',\circ,^\smile,\{e\}\rangle$, where $X\circ Y=\{x*y:x\in X,y\in Y\}$ and $X^\smile=\{x^{-1}:x\in X\}$ for $X,Y\subseteq G$.

### Basic results

Every nontrivial integral relation algebra is simple.

Every simple commutative relation algebra is integral.

Every group relation algebra is integral.

### Properties

Classtype | universal |
---|---|

Equational theory | undecidable |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | no |

Congruence distributive | yes |

Congruence modular | yes |

Congruence $n$-permutable | yes |

Congruence regular | yes |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

$n$ | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |
---|---|---|---|---|---|---|---|---|---|

# of algs | 1 | 1 | 2 | 10 | 102 | 4412 | 4886349 | 344809166311 |

For $n\ne 2^k$, the # of algebras is 0.

See http://www1.chapman.edu/~jipsen/gap/ramaddux.html for more information.