Positive Varieties of Lattice Languages

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.

💡 Research Summary

The paper “Positive Varieties of Lattice Languages” extends the algebraic theory of regular languages to the setting where each word is assigned a value from a fixed lattice Λ rather than a Boolean truth value. The authors introduce the notion of a Λ‑language, i.e., a mapping L : Σ* → Λ, and observe that the lattice operations join (∨) and meet (∧) provide natural generalizations of union and intersection for such languages. Because a complement operation is generally unavailable in a lattice, the classical notion of a variety (closed under Boolean operations, quotients, and inverse homomorphisms) cannot be directly applied. Instead, they adopt Pin’s concept of a positive variety, which requires closure under joins, meets, left and right quotients, inverse homomorphisms, and additionally under Λ‑morphisms (order‑preserving endomorphisms of the lattice).

A central technical tool is the ordered monoid (M, ≤), a monoid equipped with a compatible partial order. An order‑preserving coloring (op‑coloring) is a monotone map P : M → Λ. A Λ‑language L is said to be recognized by a triple (η, M, P) when η : Σ* → M is a monoid homomorphism and L = P ∘ η. This definition generalizes the usual acceptance condition of automata: instead of a set of accepting states, we have a lattice‑valued “coloring” of the monoid elements.

The authors prove that the class of regular Λ‑languages (those recognized by finite ordered monoids) is closed under all the operations required for a positive variety. Specifically:

1. Joins and meets of two regular Λ‑languages are recognized by the direct product of their recognizing ordered monoids, with the coloring given pointwise by the lattice join or meet of the component colorings.
2. Left and right quotients by a word u are recognized by the same monoid, with the coloring shifted by the element η(u) on the left or right.
3. Inverse homomorphisms and Λ‑morphisms preserve recognizability in a similar fashion.

These closure properties are formalized in Theorems 1 and 2, which also handle the case where a language is recognized by a divisor (submonoid or quotient) of a given ordered monoid.

The core contribution is an Eilenberg‑type correspondence between positive varieties of regular Λ‑languages and pseudo‑varieties of finite ordered monoids. A pseudo‑variety is a class of finite ordered monoids closed under taking divisors and finite direct products. The correspondence is established via the syntactic ordered monoid of a Λ‑language: define a preorder ⪯ₗ on Σ* by w₁ ⪯ₗ w₂ iff for all u, v ∈ Σ* we have L(uw₁v) ≤ L(uw₂v). The induced equivalence ≃ₗ yields the quotient monoid Mₗ = Σ*/≃ₗ equipped with the order inherited from ⪯ₗ. This monoid is the minimal ordered monoid recognizing L.

A subtle but crucial point is the inclusion of Λ‑morphisms in the definition of a positive variety. Without this closure, constant colorings (functions cons(λ) mapping every word to a fixed lattice element λ) would form a positive variety that collapses to the trivial pseudo‑variety consisting only of the one‑element monoid, thus destroying the one‑to‑one correspondence. By requiring closure under all Λ‑morphisms, the authors ensure that constant colorings are generated from the Boolean sub‑lattice {0, 1} and that the correspondence remains bijective.

Finally, the paper sketches an application to finite‑state Markov chains. By interpreting the transition probabilities as elements of a lattice (e.g., the interval