Translational hull of a Clifford semiring (Q2716704)

A Clifford semiring is an additively commutative and regular semiring \((S,+,\cdot)\) such that \(E^+=\{s\in S\mid s+s=s\}\) is a \(k\)-ideal of \(S\) and a lattice with respect to the operations on \(S\). Note that such a Clifford semiring is an inversive semiring in the sense that \((S,+)\) is an inverse semigroup. An inversive semiring \((S,+,\cdot)\) is called faithful if for all \(a\in S\) the implication \(ax,xa\in E^+\) for all \(x\in S\Rightarrow a\in E^+\) is satisfied. A left translation of a semiring \((S,+,\cdot)\) is a mapping \(\lambda\colon S\to S\) such that \(\lambda(x+y)=\lambda(x)+\lambda(y)\) and \(\lambda(xy)=\lambda(x)y\) for all \(x,y\in S\). A right translation \(\varrho\) is defined dually, and a bitranslation \((\lambda,\varrho)\) is a pair consisting of a left and a right translation. For any additively commutative semiring \((S,+,\cdot)\), the translational hull \((\Omega(S),+,\cdot)\) is the semiring \(\Omega(S)\) of all bitranslations with pointwise addition and multiplication.NEWLINENEWLINENEWLINEThe main results are: 1. The translational hull of a Clifford semiring is again a Clifford semiring. 2. Any faithful Clifford semiring is embeddable in its translational hull.