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.