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A binary algebra A=⟨A;∗⟩ is called a left [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 [x∗y=x–y] . A symmetric difference group is a binary algebra satisfying all the identities common to both left and right difference groups. In this note we determine the structure of a free symmetric difference group. Using this, we show that an identity is common to both left difference and right difference if and only if it is a formal consequence of a single identity. This includes the known result that the theories of left and right difference groups are one-based. As an application of this procedure, we show that any finite set of equations which includes the axioms for rings with unit is logically equivalent to a single equation. This was announced without proof by Alfred Tarski and Ralph McKenzie (see [5]). We take this opportunity to mention some new open problems. This is a joint work done in collaboration with George Grätzer.

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2. Kelly, David; Padmanabhan; Wolk, B. Identities common to addition and subtraction. Houston J. Math. 11 (1985), no. 3, 335–343.
3. Padmanabhan, R.; Penner, P. An implication basis for linear forms. Algebra Universalis 55 (2006), no. 2–3, 355–368.
4. Padmanabhan, R.; Quackenbush, R. W. Equational theories of algebras with distributive congruences. Proc. Amer. Math. Soc. 41 (1973), 373–377.
5. George F. McNulty, Minimum bases for equational theories of groups and rings: The work of Alfred Tarski and Thomas C. Green, Annals of Pure and Applied Logic, 127 (2004), 131–153.