A variety of monoids is a class of monoids closed under homomorphic images, submonoids and direct products, equivalently the class of all monoids satisfying a fixed set of identities. Ordered by inclusion these varieties form a complete lattice MON, whose meet is intersection and whose join is the variety generated by the union. The object of study is the internal structure of MON: its atoms and coatoms, its intervals and distinguished sublattices, which varieties have modular, distributive, or finite subvariety lattices, which are limit varieties or Cross varieties, and which elements are definable in MON up to automorphism. The monoid case is contrasted throughout with the lattice SEM of semigroup varieties, which behaves quite differently despite the near-identical signature.
A variety of monoids is a class of monoids closed under submonoids, homomorphic images and arbitrary direct products; by Birkhoff's theorem these are exactly the classes defined by a set of identitie…