## 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 undecidable undecidable undecidable no no yes yes yes yes yes yes no no

### Finite members

 $n$ # of algs 1 2 4 8 16 32 64 128 256 1 1 2 10 102 4412 4886349 344809166311

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