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Kleene equality

Equality operator on partial functions

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Source updated May 21, 2026

In mathematics, Kleene equality, or strong equality, ( $\simeq$ ) is an equality operator on partial functions, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal.

For example, if we have partial functions $f$ and $g$ , $f\simeq g$ means that for every $x$ :

$f(x)$ and $g(x)$ are both defined and $f(x)=g(x)$
or $f(x)$ and $g(x)$ are both undefined.

Some authors are using "quasi-equality", which is defined like this: $(y_{1}\sim y_{2}):\Leftrightarrow ((y_{1}\downarrow \lor y_{2}\downarrow )\longrightarrow y_{1}=y_{2}),$ where the down arrow means that the term on the left side of it is defined. Then it becomes possible to define the strong equality in the following way: $(f\simeq g):\Leftrightarrow (\forall x.(f(x)\sim g(x))).$

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This article is adapted from the Wikipedia article “Kleene equality”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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