On Algebras all of Whose Subalgebras are Simple; Some Solutions of Plonka's Problem

Sin-Min Lee

For each cardinal number $\alpha\geq 1$, we construct two types of grupoids $\langle X_\alpha;\circ\rangle$ and $\langle X_\alpha; *\rangle$ which are hereditarily simple and have subgrupoids of all small orded. If $\alpha\geq \aleph_0$, we show that they both admit only discrete topology to become topological grupoids. An application of the grupoid $\langle X_\alpha; *\rangle$ in the theory of non-associative rings is indicated.