Department of Mathematics
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Consider the two familiar binary algebras: (Z; +, 0) and (Z; –, 0). While the binary addition is both commutative and associative, the binary subtraction is neither commutative nor associative. In this talk, we show the technique of finding all identities common to these two operations. The idea of congruences (universal analogs of "normal subgroups" in group theory and "ideals" in ring theory) are employed in discovering these equations. Using these tools, we prove that a groupoid identity of type (2, 0) is universally valid for both addition and subtraction in abelian groups if and only if it is a formal consequence of the finite set

Similar theorems are known for other familiar operations in abelian groups and in commutative Moufang loops. Such a characterization is not known for groups or loops in general. These results are taken from a series of papers written in collaboration with George Gratzer, David Kelly, William McCune and Barry Wolk.

REFERENCES

• Grätzer, G.; Padmanabhan, R. Symmetric difference in abelian groups. Pacific J. Math. 74 (1978), 339–347.
• Kelly, David; Padmanabhan, R.; Wolk, B. Identities common to addition and subtraction. Houston J. Math. 11 (1985), no. 3, 335–343.
• Kelly, David; Padmanabhan, R. Identities common to four abelian group operations with zero. Algebra Universalis 21 (1985), 1–24.
• McCune, W; Padmanabhan, R. Symmetric difference in commutative Moufang loops (to appear).