Explore the algebraic structures of semigroups and semilattices in functional programming during this 21-minute conference talk from YOW! 2016. Delve into how these mathematical concepts, related to monoids, can be applied by programmers. Learn about their uses, familiar concretions, and the trade-offs involved in choosing the right algebraic structure for specific programming tasks. Examine examples using Haskell, including numbers under addition, functions, and foldable structures like lists and Maybe types. Discover the importance of laws in these structures and how they relate to real-world applications such as CRDTs (Conflict-free Replicated Data Types). Gain insights into promoting semigroups, understanding idempotence, and working with bounded join-semilattices, using practical examples like set union laws.