Positive Varieties of Lattice Languages

TL;DR

This paper extends the theory of positive varieties to lattice languages, establishing a correspondence with pseudo-varieties of finite ordered monoids and exploring algebraic applications to Markov chains.

Contribution

It generalizes Pin's positive variety theorem to lattice languages and introduces an algebraic framework for finite-state Markov chains.

Findings

Established a one-to-one correspondence between positive varieties of regular lattice languages and pseudo-varieties of finite ordered monoids.

Extended algebraic methods to analyze finite-state Markov chains within the lattice language framework.

Abstract

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.