On the semigroup structure of cyclic left distributive algebras (Q1895803)

An algebra \(G(\circ, *)\) is called left distributive if it satisfies the identities \((a \circ b) \circ c = a \circ (b \circ c)\), \((a \circ b) * c = a * (b * c)\), \(a \circ b = (a * b) \circ a\), \(a * (b \circ c) = (a * b) \circ (a * c)\). It follows that the operation \(*\) is left distributive: \(a * (b * c) = (a * b) * (a * c)\). For every \(n > 0\) there exists just one left distributive operation \(*\) on the set \(P_n = \{0, 1, \dots, 2^n - 1\}\) such that \(a * 1 = (a + 1) \text{ mod } 2^n\); the operation \(\circ\) may be defined on \(P_n\) by \((a \circ b) * 1 = a * (b * 1)\). Idempotents, regular elements and generators of the semigroup \((P_n, \circ)\) are described; e.g. \((P_n, \circ)\) is generated by the set \(R_n\) of rare elements, i.e. elements \(a \in P_n\) such that \(a = b * c\) implies \(b= 0\) or \(b * 1 = a\) for any \(b, c \in P_n\).