# What does it mean for a relation algebra to be simple?

I am reading "Boolean Algebras with Operators part II" by Bjarni Jonsson and Alfred Tarski. On theorem 4.10 (p.132-133), they refer to a relation algebra $\mathfrak{A}$ being "simple" and proves that it is equivalent to $\mathfrak{A}$ having no ideal elements different from 0 and 1. What definition of "simple" is being used in this context? The article is in JSTOR, by the way.

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## 1 Answer

According to the definitions given in the extended abstract available here, a simple relation algebra is one that satisfies the condition $1;r;1=1$ for every non-zero $r$, where ‘;’ is the composition operator. Apparently an ideal element is an element $r$ that satisfies $1;r;1=r$.

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That can't be it his definition, since in theorem 4.10 he proves that definition you gave is equivalent to $\mathfrak{A}$ being simple. –  Alison Oct 5 '11 at 5:47