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.
Definition of a pregroup A pregroup is a partially ordered algebra ( A , 1 , ⋅ , − l , − r , ≤ ) {\displaystyle (A,1,\cdot ,-^{l},-^{r},\leq )} such that ( A , 1 , ⋅ ) {\displaystyle (A,1,\cdot )} is a monoid, satisfying the following relations:
x l ⋅ x ≤ 1 x ⋅ x r ≤ 1 {\displaystyle x^{l}\cdot x\leq 1\qquad x\cdot x^{r}\leq 1} (contraction)
1 ≤ x ⋅ x l 1 ≤ x r ⋅ x {\displaystyle 1\leq x\cdot x^{l}\qquad 1\leq x^{r}\cdot x} (expansion) The contraction and expansion relations are sometimes called Ajdukiewicz laws. From this, it can be proven that the following equations hold:
1 l = 1 = 1 r {\displaystyle 1^{l}=1=1^{r}}
x l r = x = x r l {\displaystyle x^{lr}=x=x^{rl}}
( x ⋅ y ) l = y l ⋅ x l ( x ⋅ y ) r = y r ⋅ x r {\displaystyle (x\cdot y)^{l}=y^{l}\cdot x^{l}\qquad (x\cdot y)^{r}=y^{r}\cdot x^{r}}
x l {\displaystyle x^{l}} and x r {\displaystyle x^{r}} are called the left and right adjoints of x, respectively. The symbols ⋅ {\displaystyle \cdot } and ≤ {\displaystyle \leq } are also written ⊗ {\displaystyle \otimes } and → {\displaystyle \to } respectively. In category theory, pregroups are also known as autonomous categories or (non-symmetric) compact closed categories. More typically, x ⋅ y {\displaystyle x\cdot y} will just be represented by adjacency, i.e. as x y {\displaystyle xy} .
Definition of a pregroup grammar A pregroup grammar consists of a lexicon of words (and possibly morphemes) L, a set of atomic types T which freely generates a pregroup, and a relation : {\displaystyle :} that relates words to types. In simple pregroup grammars, typing is a function that maps words to only one type each.
Examples Some simple, intuitive examples using English as the language to model demonstrate the core principles behind pregroups and their use in linguistic domains. Let L = {John, Mary, the, dog, cat, met, barked, at}, let T = {N, S, N0}, and let the following typing relation hold:
John : N Mary : N the : N ⋅ N 0 l dog : N 0 cat : N 0 {\displaystyle {\textit {John}}:N\qquad {\textit {Mary}}:N\qquad {\textit {the}}:N\cdot N_{0}^{l}\qquad {\textit {dog}}:N_{0}\qquad {\textit {cat}}:N_{0}}
met : N r ⋅ S ⋅ N l barked : N r ⋅ S at : S r ⋅ N r r ⋅ N r ⋅ S ⋅ N l {\displaystyle {\textit {met}}:N^{r}\cdot S\cdot N^{l}\qquad {\textit {barked}}:N^{r}\cdot S\qquad {\textit {at}}:S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l}}
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