# What does Tarski mean by a “tautological operation” on a Boolean algebra?

I am reading Part II of Chin and Tarski's "Distributive and Modular Laws in the Arithmetic of Relation Algebras". In the beginning of section 4, the authors say "In general, if $\odot$ is a binary operation which is commutative, associative, and tautological, then both the left and right distributive laws for $\odot$ under $\odot$ are identically satisfied". My question is: what does "tautological" mean in this context?

We are working within a fixed Boolean algebra $B$. $\odot$ is some binary operation on the universe of $B$. He is saying that if $\odot$ is commutative, associative, and "tautological", then $\odot$ is left and right distributive over $\odot$, i.e., for all $x$, $y$, and $z$ in the universe of $B$, $x\odot (y\odot z)=(x\odot y)\odot (x\odot z)$ and $(y\odot z)\odot x=(y\odot x)\odot (z\odot x)$. I can't find the word "tautological" (in reference to a binary relation) used anywhere else.

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Do they use the term anywhere else?  Can you give us any more of the text, or context here? – Doug Spoonwood Mar 27 '12 at 1:31
Could it possibly mean idempotent? – Miha Habič Mar 27 '12 at 5:44

## 1 Answer

Yes, it means idempotence (for binary operations). A binary operation $\odot$ is tautological iff $\forall x\ [x \odot x = x]$. Perhaps the name comes from the fact that conjunction and disjunction satisfy this tautologically. I think this is old terminology, but I personally find it somewhat appealing to reserve idempotence for unary operations since the idea is somewhat different. Anyway, here's an example, where it's called the law of tautology (pg. 211):

Frink, Orrin. New Algebras of Logic. The American Mathematical Monthly , Vol. 45, No. 4 (Apr., 1938), pp. 210-219 http://www.jstor.org/stable/2302605

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Thank you, Rachel :). – MathMastersStudent Mar 28 '12 at 2:11
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