We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\widehat{\mathbb{Z}}$ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with $\widehat{\mathbb{Z}}$-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.

我们用各种无扭系数对联结 K 理论中的加性运算进行分类。我们发现积分情况的答案需要理解$\widehat{\mathbb{Z}}$案件。此外，虽然积分加法运算是由 Adams 运算在拓扑上生成的，但它们并没有简化为后者的无限线性组合。我们描述了稳定运行的拓扑基础，并将其与分级 K 理论中的稳定运行基础联系起来。我们对两种理论中的乘法运算进行分类，并表明齐次加法稳定运算具有$\widehat{\mathbb{Z}}$-系数是通过稳定的乘法运算拓扑生成的。对于积分运算则不然。