While a language assigns a value of either `yes' or `no' to each word, a lattice language assigns an element of a given lattice to each word. An advantage of lattice languages is that joins and meets of languages can be defined as generalizations of unions and intersections. This fact also allows for the definition of positive varieties -- classes closed under joins, meets, quotients, and inverse homomorphisms -- of lattice languages. In this paper, we extend Pin's positive variety theorem, proving a one-to-one correspondence between positive varieties of regular lattice languages and pseudo-varieties of finite ordered monoids. Additionally, we briefly explore algebraic approaches to finite-state Markov chains as an application of our framework.

Accepted to DLT 2025