Vol. 37(51), pp. 33--36 (1985)
On algebras all of whose subalgebras are simple; some solutions of Plonka's problem
Sin-Min LeeDepartment of Mathematics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
Abstract: 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.
Classification (MSC2000): 20L05; 17E05
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