S-系的扭类

TORSION CLASSES OF S-SYSTEMS

假设S是有0,1的半群,τ是S-系的遗传扭论,对任意的右S-系M,T_τ(M)是M的τ-扭根。x∈T_τ(M)当且仅当存在某个τ-稠密右同余ρ,使得对任意的(S_1,S_2)∈ρ均有xs_1=xs_2,同时,当右S-系M是τ-扭自由时,M的τ-稠密同余是M的本质同余,特别,对忠实的遗传扭论τ,S的τ-稠密右同余是S的本质右同余。

Let S be a semigroup with O and 1, τ be a hereditary torsion theory on S-systems, T_τ(M) be τ-torsion radical of S-system M. x∈T_τ(M) if and only if XS_1=XS_2 for some τ-dense right congruence ρ on S and A (S_1, S_2)∈ρ. If M is τ-torsion free, then τ-dense congruences of M is essential. In particular, if τ is a faithfal hereditary torsion theory, then τ-dense right congruences of S is essential.