pregroup

A pregroup is an algebraic structure used mainly in linguistics and logic to model how elements can combine and cancel each other. Formally, it is a partially ordered monoid (P, ≤, ·, 1) equipped with a pair of operations providing left and right adjoints: for each element a in P there are elements a^l and a^r. The partial order is compatible with multiplication, and the adjoints satisfy contraction rules that allow reductions when composing elements.

The key feature of a pregroup is the ability to reduce sequences of elements using the adjoints. In typical formulations, the contractions a^l a ≤ 1 and a a^r ≤ 1 hold, enabling pairs of adjacent types to cancel against a distinguished unit or target type. These reductions model how linguistic components combine to form well-formed phrases. The structure also supports expansions, ensuring that relations between types behave coherently under composition.

In practice, pregroups are used to assign types to words in a sentence and to determine grammaticality by reducing the sequence of types to a desired target type. A common example uses a small set of base types, such as n for noun and s for sentence. A transitive verb has the pregroup type n^r s n^l. In a sentence like “Alice loves Bob,” with the sequence n · (n^r s n^l) · n, contractions n n^r → 1 and n^l n → 1 reduce the string to s, indicating a well-formed sentence.

Pregroups were introduced by Joachim Lambek as a tool for formal grammar and have influenced developments in categorial grammar and type-logical linguistics. They provide a compact algebraic framework for reasoning about word order and grammatical structure.