\(\Gamma\)-semirings. II
[Part I, ibid. 19, No. 1, 49-54 (1995; Zbl 0846.16034).] Let \(M\) and \(\Gamma\) be additive semigroups. If there exists a mapping \(M\times\Gamma\times M\to M\) (images to be denoted \(x\alpha y\), \(x,y\in M\), \(\alpha\in \Gamma\)) satisfying, for all \(x,y,z\in M\), \(\alpha,\beta\in\Gamma:\) (i) \(x\alpha(y\beta z)=(x\alpha y)\beta z\); (ii) \(x\alpha(y+z)=x\alpha y+x\alpha z\); \(x(\alpha+\beta)y=x\alpha y+x\beta y\); \((x+y)\alpha z=x\alpha z+y\alpha z\), then \(M\) is called a \(\Gamma\)-semiring. If for each \(x\in M\), there exists a unique \(x'\in M\) such that \(x=x+x'+x\) and \(x'=x'+x+x'\), then \(M\) is called an additively inverse \(\Gamma\)-semiring. In this paper, various results are proved for additively inverse \(\Gamma\)-semirings which correspond to similar results for semirings.
scientific article; zbMATH DE number 1267315