Pregroup grammar

Pregroup grammar (PG) is a grammar formalism intimately related to categorial grammars. Much like categorial grammar (CG), PG is a kind of type logical grammar. Unlike CG, however, PG does not have a distinguished function type. Rather, PG uses inverse types combined with its monoidal operation.

A pregroup is a partially ordered algebra ${\displaystyle (A,1,\cdot ,-^{l},-^{r},\leq )}$ such that ${\displaystyle (A,1,\cdot )}$ is a monoid, satisfying the following relations:

The contraction and expansion relations are sometimes called Ajdukiewicz laws.

From this, it can be proven that the following equations hold:

${\displaystyle x^{l}}$ and ${\displaystyle x^{r}}$ are called the left and right adjoints of x, respectively.

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