An Application of Lattice Theory to Quasigroups
نویسندگان
چکیده
The purpose of this note is to show that O. Ore's general formulation of the Jordan-Holder theorems in partially ordered sets [3] yields the Jordan-Holder theorems for loops [ l ] which have recently been obtained by A. A. Albert [2]. We shall assume that the reader is familiar with the papers [l, 2, 3] of O. Ore and of A. A. Albert. We begin with an examination of certain congruence relations in loops. We are led to characterize the normal divisors of A. A. Albert as those subloops which commute and associate with the elements of the loop. Our proof of the fundamental quadrilateral condition of O. Ore [3] is then based on this characterization. A loop ® is a quasigroup with an identity element e. For each nonnull subset § C ® we define a relation Hp on ®® as follows:
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