We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval $[d,1]$ to be a subalgebra of an involutive residuated lattice, where $d$ is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

我们证明了 MV 代数和 MV 半环之间的等价项提升到对合剩余格和一类称为对合半环的半环。半环视角导致区间的一个充要条件$[d,1]$是一个对合剩余晶格的子代数，其中$d$是二元化元素。我们还在这类半环的研究中引入了一些半模理论的成果和技术，概括了单射和射影MV-半模的结果。事实上，我们注意到对合起着至关重要的作用，并且只要不涉及 Mundici 函子，对合半环的 MV 半环的结果仍然是正确的。特别是，我们证明了对合是射影半模和单射半模重合的充分必要条件。