**Lattice Theory for Parallel Programming** *Course diary 2024 for the lattice theory part* # Ressources The course material for the lattice theory part is available in this [shared folder](https://uniluxembourg-my.sharepoint.com/:f:/g/personal/bruno_teheux_uni_lu/EtN22uWEUjRNkC0JlL6ftB4Bfs_HOX17ewSzYFkUaxk14A?e=1flrfv). The following introductory textbook can be consulted for further reference. An electronic version is available for free via [a-z.lu](https://www.a-z.lu). [#1]: B. Davey and H.A. Priestley. *Introduction to lattice and order*. Second edition. Cambridge University Press, 2002. # Week 1: Sepember 20, 2024 **Topics covered:** Order and poset. Constructions of partial orders. Mappings between orders. Special subsets in posets. # Week 2: September 27, 2024. **Topics covered:** Bounds, lattices, complete lattices: definitions and examples. Lattices as algebras: definition and correspondence with lattices as posets. # Week 3: October 11, 2024. **Topics Covered:** Maps between lattices: (bounded) homomorphisms, isomorphisms and link with monotone map. Sublattices. Homomorphic images. Class operators $\mathbb{H}$, $\mathbb{S}$ $\mathbb{P}$, terms, equations, equational classes and the $\mathbb{HSP}$ theorem. # Week 4: October 18, 2024. **Topics Covered:** Poset of partial maps. Principal ideals in this poset are complete lattices. Top $\bigcap$-structures. Order isomorphisms between complete lattices preserve all joins and all meets. Alternative characterization of complete lattices. A lattice without infinite chain is complete (without proof). Fixpoints and existence of greatest and least fixpoints in a complete lattice. Definition of a closure operator. *Remark*. We have delayed the section about complements and Boolean Algebras. # Week 5: October 30, 2024. **Topics covered:** Closure operators and complete lattices. Galois connection: definition and first example. # Week 6: November 15, 2024. **Topics covererd:** Galois connections: definition, examples (including interval lattices), alternative characterizations. Galois embedding: definition, example, alternative charaterization. Construction of Galois connections from exising ones: dual order, compositions, function lifting.