On the structure of algebraic systems
Tsuyoshi Fujiwara · Proceedings of the Japan Academy Series A Mathematical Sciences · 1954
The structure of an algebraic system A has been discussed by K. Shoda under the iollowing conditions: SI.A has a null-element.SII.The subsystem generated by any two normal subsystems of A is normal in A.SIII.The meromorphism of any two algebraic systems which are homomorphic to A is aways class.meromorphism.(SII and SIII are assumed for any subsystem of A.) G. Birkhoff has introduced in his book ) the following condition which is equivalent to SIII" all congruences on A are permutable.In the present paper we shall give a new definition of normal subsystems, and study on the normal subsystems and the congruences of an algebraic system A ( 1).Moreover under weaker conditions than SII, SIII ( 2), we shall discuss the Jordan-HSlder-Schreier theorem ( 3) and the Remak-Schmidt-Ore theorem for A (4).