Varieties

This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and motivate the fundamental concepts. We do this for the most part without proofs, and often without precise definitions, leaving these to the formal development of the theory that begins in Section 2. Our presentation of the theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the equational theory of lattices of regular languages. In the subsequent sections we consider in more detail aspects of varieties that were only briefly evoked in the introduction: Decidability, operations on languages, and characterizations in formal logic.

💡 Research Summary

The paper “Varieties” by Howard Straubing and Pascal Weil presents a comprehensive treatment of the algebraic theory of varieties as a tool for classifying regular languages. Drawing heavily on the equational approach of Gehrke, Grigorieff and Pin (2008), the authors first motivate the main concepts through a series of intuitive examples before moving to a formal development.

The first substantive example concerns languages whose membership depends only on the set of letters occurring in a word. Such languages are exactly those recognized by the two‑element monoid U₁ = {0,1} (or its direct powers). The corresponding pseudovariety, denoted J₁ (also called S₁), consists of all finite monoids that are both idempotent and commutative. This class can be described by the identities xy = yx and x² = x. Because these identities can be checked in polynomial time on a multiplication table, membership of a regular language in J₁ is decidable: compute the syntactic monoid Synt(L) and verify the two identities.

The second major family studied is the piecewise‑testable languages. A word v = a₁…aₖ is a subword of w if w can be written with the letters of v appearing in order, possibly with other letters in between. The basic languages L_v = A* a₁ A* … aₖ A* generate a Boolean algebra; the Boolean closure of these languages yields the piecewise‑testable class. Algebraically, this class coincides with the pseudovariety J of J‑trivial monoids (monoids whose Green’s J‑relation is trivial). Unlike J₁, J cannot be captured by ordinary identities; instead one needs profinite identities involving the ω‑power, e.g. (xy)^ω = (yx)^ω and xx^ω = x^ω (or equivalently (xy)^ω x = y (xy)^ω). These identities are preserved under finite direct products and quotients, thereby defining a pseudovariety. Consequently, deciding piecewise‑testability reduces to checking whether Synt(L) satisfies the profinite identities, which can be done efficiently from the multiplication table.

Logical characterizations accompany both algebraic classes. For J₁, the authors consider a first‑order logic over word positions with unary predicates Q_a(x) meaning “the letter at position x is a”. Sentences built from such predicates (without equality) define exactly the languages whose membership depends only on the set of letters; thus J₁ = FO