Vol. 74, No. 2, 1978

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Symmetric difference in abelian groups

George Grätzer and R. Padmanabhan

Vol. 74 (1978), No. 2, 339–347

A groupoid A = A;∗⟩ is called a left (resp. right) difference group if there is a binary operation + in A such that the system A;+is an abelian group and x y = x + y (resp. x y = x y). A symmetric difference group is a groupoid satisfying all the identities common to both left and right difference groups. In this note we determine the structure of a symmetric difference group. Using this, we show that any finitely based equational theory of symmetric difference groups is one-based. This includes the known result that the theories of left and right difference groups are one-based. Other known results on finitely based theories of rings also follow.

Mathematical Subject Classification 2000
Primary: 20K99
Received: 26 March 1974
Revised: 10 September 1977
Published: 1 February 1978
George Grätzer
R. Padmanabhan