二元集的16种代数运算

Sixteen Algebraic Operations on Binary Set

用抽象代数的方法系统分析二元集上仅有的16种运算,指出这16种运算中只有异或运算和等值运算能对二元集构成群;异或运算和与运算能使二元集成为环以及等值运算和或运算能使二元集成为环;异或运算可以推广到扩展的自然数集,并使扩展的自然数集成为群。并给出扩展的自然数集上异或运算的一个性质定理。

Sixteen algebraic operations which binary set has only are systematically analysed with abstract algebra mathods. It is pointed out that out of the sixteen operations only exclusive disjunctional operation and bi-conditional operation can make binary set form group; exclusive disjunctional operation and 'conjunctional operation can make binary set form ring ; bi-conditional operation and disjunctional operation can also make binary set form ring; exclusive disjunctional operation can be generalized on extensive natural set and make it form group. Meanwhile, a property theorem of exclusive disjuctional operation on extensive natural set is also given.