# Dualizing sup-preserving endomaps of a complete lattice

@inproceedings{Santocanale2020DualizingSE, title={Dualizing sup-preserving endomaps of a complete lattice}, author={Luigi Santocanale}, booktitle={ACT}, year={2020} }

It is argued in [5] that the quantale [L,L]∨ of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in [16] that this Girard quantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney’s transforms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category of complete lattices and sup-preserving maps) whose objects are the completely distributive lattices…

## 2 Citations

### Frobenius structures in star-autonomous categories

- MathematicsArXiv
- 2022

. It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is com- pletely distributive. Since completely distributive…

### Unitless Frobenius quantales

- MathematicsArXiv
- 2022

. It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can deﬁne Frobenius quantales that may not have a unit. We develop the elementary…

## References

SHOWING 1-10 OF 22 REFERENCES

### Tight Galois connections and complete distributivity

- Mathematics
- 1960

and discusses its relations with the property of complete distributivity in complete lattices. A procedure for constructing Galois connections between complete lattices is presented. The Galois…

### Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories.

- Philosophy
- 1997

This note applies techniques we h a ve developed to study coherence in monoidal categories with two tensors, corresponding to the tensor{par fragment o f linear logic, to several new situations,…

### The structure of Galois connections.

- Mathematics
- 1974

This paper deals with Galois connections between two partially ordered sets (posets) A, B. The first sections are devoted to the construction of all Galois connections between A and B. The last…

### Residuated lattices: An algebraic glimpse at sub-structural logics

- Mathematics
- 2007

This book considers both the algebraic and logical perspective within a common framework, and shows how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones.

### A New Lattice Construction: The Box Product

- Mathematics
- 1999

In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisﬁes a very…