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===First Kleene algebra===
The first Kleene algebra <math>\mathcal{K}_1</math> consists of the set <math>\mathbb{N}</math> with application <math>a b := \phi_a(b)</math>, where <math>\phi_a</math> denotes the <math>a</math>-th [[partial recursive function]] in a standard Gödel numbering.{{r|van-oosten|p=15}}{{r|bauer|p=29}}
This pca can also be [[relativization|relativized]] to an oracle <math>D \subseteq \mathbb{N}</math>: we define a pca <math>\mathcal{K}_1^D</math> with carrier <math>\mathbb{N}</math> by setting <math>a b := \phi^D_a(b)</math>, where <math>\phi^D_a</math> is the <math>a</math>-th partial recursive function with oracle <math>D</math>.{{r|van-oosten|p=15}}{{r|bauer|p=30}}
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