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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.


This is a pre-copy-editing, author-produced PDF of an article that underwent peer review and was accepted for publication in Journal of Algebra and Its Applications in 2021. This article may not exactly replicate the final published version. The definitive publisher-authenticated version is available online at

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World Scientific Publishing Company

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Algebra Commons



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