Finite left distributive algebras with one generator Original Research Article

Finite monogenerated groupoids G satisfying the left distributive law x · (y · z) = (x · y) · (x · z) are studied. They are shown to reduce over image and image to a groupoid isomorphic to Ak = Ak(*), k ≥ 0. (Ak is the unique left distributive groupoid on {1, …, 2k} with a * 1 triple bond; length as m-dash a + 1 mod 2k for every 1 ≤ a ≤ 2k.) G congruent with Ak is proved to hold whenever b → a · b equals idG for some a ε G. We describe all cases when G = Ga union or logical sum {b} for some a, b ε G, and all cases when there exists a binary operation o on G such that G(·, o) satisfies the axioms of left distributive algebras.