Extended residuated lattices

In the paper, we investigate algebraic extensions of residuated lattices and additively idempotent commutative semirings. First, based on the definition of EMV-algebras, we introduce the notion of extended residuated lattices by using the generalized Boolean center, in such a way that every extended residuated lattice contains a residuated lattice. We prove that every extended residuated lattice with a top element is termwise equivalent to a residuated lattice. Also, we show some relations between extended residuated lattices and EMV-algebras. And we prove that every EMV-algebra is termwise equivalent to an extended regular and divisible residuated lattice. In particular, the category of EMV-algebras is a reflective subcategory of the category of extended divisible residuated lattices. Moreover, based on the definition of EMV-semirings, we introduce extended pseudocomplemented semirings and investigate two subclasses of extended pseudocomplemented semirings, which are extended involutive semirings and extended Stone semirings. We show that an extended involutive semiring can be organized into an extended regular residuated lattice. Conversely, every extended regular residuated lattice can be considered as an extended involutive semiring. In particular, we get that the categories of extended regular residuated lattices and extended involutive semirings are isomorphic. Finally, we show that an extended pseudocomplemented semiring is Stone if and only if the skeleton of it can form a generalized Boolean algebra. Also, we obtain the relationship between extended Stone semirings and extended Stonean residuated lattices.