**Lattice Theory for Parallel Programming** *Course diary 2025 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/Eook7fUmKz5DhXezqK9MfZsBUW5vggO5itISwWw44IDr-w?e=WRQKGl). 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: September 24, 2025 **Topics covered:** Order and poset. Constructions of partial orders. Mappings between orders. Special subsets in posets. # Week 2. October 8, 2025 **Topics covered:** Bounds, lattices, complete lattices: definitions and examples. Fix points: definition, examples, fix points in finite lattices, Tarski's fix point theorem. Poset of partial maps: definition, and meet and join operations. # Week 3. October 15, 2025 **Topics Covered:** Ideal generared by a total function. Constant propagation. Galois connection: définition, first properties and exemples (included the lattice of intervals.) Lattices as algebras: definition and correspondence with lattices as posets. Maps between lattices: (bounded) homomorphisms, isomorphisms and link with monotone map. Sublattices. # Week 4. November 5, 2025 **Topics Covered:** Bounded sublattices. Lattice embeddings. Homomorphic images. Class operators $\mathbb{H}$, $\mathbb{S}$ $\mathbb{P}$, terms, equations, equational classes and the $\mathbb{HSP}$ theorem. Complemented lattices and Boolean algebras. Complete lattices: alternative definitions. Lattice isomorphisms between complete lattices. # Week 5. November 19, 2025 **Topics covered:** Closure operators and complete lattices. Galois connection: definition and first example. From closure operators to Galois connections. From Galois connections to closure operators. # Week 6. December 5, 2025 **Topics covererd:** Galois connection and preservations of joins and meets. Existence of upper ajoints. Galois connections on complete lattices. Galois embeddings.