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I am doing some research on the algebra of relations (Tarski/Givant axiomatization), and I notice that in a proper relation algebra, an element (relation) is an equivalence element if and only if it is transitive and symmetric. I am wondering as to why "reflexive" is not included. Is there even a possible sentence or set of sentences in the language of relation algebras such that in a proper relation algebra, an element is reflexive if and only if it satisfies it (the sentence or set of sentences)? I am trying to figure out if there is a way to formalize reflexivity in the theory of relation algebras. It would be preferable if there were an equational axiom that could capture reflexivity.

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 If you have access to the symbols, $x \cap I = I$. $I$ is the unique element that satisfies $\forall y,I\cdot y=y$. If not, you could try $\forall y\neq 0,y \cdot x \cdot y^T\neq 0$. This is part of the general, sort of category-theoretic trick that an element of a set is just a kind of subset, and so a statement about all elements might really be a statement about all subsets. – Will Sawin Nov 18 2011 at 8:04

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