Weighted (b,c)-core inverses in semigroups with involution
Jingjing Song, Huihui Zhu, Honglin Zou · Filomat · 2024
Let S be a +-moniod and let a, b, c, v, w ? S. In this paper, we define the (v, w)-weighted (b, c)-core inverse of a. The element a is called (v, w)-weighted (b, c)-core invertible if there exists x ? S such that xvcawx = x, xvS = bS and Swx = Sc+. It is shown that the core inverse, the w-core inverse and the (b,c)-core inverse are special cases of the defined (v,w)-weighted (b,c)-core inverse. Several criteria for the (e,w)-weighted (b,c)-core inverse are given, where e is an invertible Hermitian element. For instance, it is proved that a is (e,w)-weighted (b,c)-core invertible if and only if there exists some x ? bS such that xecawb = b, cawxec = c and (cawx)+ = cawx if and only if aw is (b, c)-invertible and c (ca or cawb) is {e, 1, 3}-invertible. The dual (v, w)-weighted (b, c)-core inverse of a is defined by the existence of y ? S satisfying yvabwy = y, yvS = b+S and Swy = Sc. Dual results for the dual (v, w)-weighted (b, c)-core inverse are also established. Finally, when S is a unital +-ring, the (dual) weighted (b, c)-core inverse is characterized by the direct sum.