(Pre-)Algebras for Linguistics 4. Residuation Carl Pollard Linguistics 680: Formal Foundations Autumn 2010 Carl Pollard (Pre-)Algebras for Linguistics
Inflationary and Deflationary Operations A unary operation f on a preorder � P, ⊑� is called inflationary iff, for all p ∈ P , p ⊑ f ( p ) Carl Pollard (Pre-)Algebras for Linguistics
Inflationary and Deflationary Operations A unary operation f on a preorder � P, ⊑� is called inflationary iff, for all p ∈ P , p ⊑ f ( p ) deflationary iff, for all p ∈ P , f ( p ) ⊑ p Carl Pollard (Pre-)Algebras for Linguistics
Closure and Interior Operations Suppose � P, ⊑� is a preorder, and f : P → P a unary operation which is both monotonic and idempotent u.t.e. Then f is called a closure operation if it is inflationary Carl Pollard (Pre-)Algebras for Linguistics
Closure and Interior Operations Suppose � P, ⊑� is a preorder, and f : P → P a unary operation which is both monotonic and idempotent u.t.e. Then f is called a closure operation if it is inflationary an interior (or kernel ) operation if it is deflationary Carl Pollard (Pre-)Algebras for Linguistics
Closure and Interior Operations Suppose � P, ⊑� is a preorder, and f : P → P a unary operation which is both monotonic and idempotent u.t.e. Then f is called a closure operation if it is inflationary an interior (or kernel ) operation if it is deflationary Examples topology of the real line Carl Pollard (Pre-)Algebras for Linguistics
Closure and Interior Operations Suppose � P, ⊑� is a preorder, and f : P → P a unary operation which is both monotonic and idempotent u.t.e. Then f is called a closure operation if it is inflationary an interior (or kernel ) operation if it is deflationary Examples topology of the real line modal operators Carl Pollard (Pre-)Algebras for Linguistics
Closure and Interior Operations Suppose � P, ⊑� is a preorder, and f : P → P a unary operation which is both monotonic and idempotent u.t.e. Then f is called a closure operation if it is inflationary an interior (or kernel ) operation if it is deflationary Examples topology of the real line modal operators Kleene closure operation on languages Carl Pollard (Pre-)Algebras for Linguistics
Residuated Pairs Suppose � P, ⊑� and � Q, ≤� are preorders and f : P → Q , g : Q → P functions such that, for all p ∈ P, q ∈ Q , f ( p ) ≤ q iff p ⊑ g ( q ) Carl Pollard (Pre-)Algebras for Linguistics
Residuated Pairs Suppose � P, ⊑� and � Q, ≤� are preorders and f : P → Q , g : Q → P functions such that, for all p ∈ P, q ∈ Q , f ( p ) ≤ q iff p ⊑ g ( q ) Then we say: � f, g � is a residuated pair Carl Pollard (Pre-)Algebras for Linguistics
Residuated Pairs Suppose � P, ⊑� and � Q, ≤� are preorders and f : P → Q , g : Q → P functions such that, for all p ∈ P, q ∈ Q , f ( p ) ≤ q iff p ⊑ g ( q ) Then we say: � f, g � is a residuated pair g is a residual of f Carl Pollard (Pre-)Algebras for Linguistics
Residuated Pairs Suppose � P, ⊑� and � Q, ≤� are preorders and f : P → Q , g : Q → P functions such that, for all p ∈ P, q ∈ Q , f ( p ) ≤ q iff p ⊑ g ( q ) Then we say: � f, g � is a residuated pair g is a residual of f f is a residuation of g Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: both f and g are monotonic Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: both f and g are monotonic g ( q ) is a greatest element of { p ∈ P | f ( p ) ≤ q } Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: both f and g are monotonic g ( q ) is a greatest element of { p ∈ P | f ( p ) ≤ q } f ( p ) is a least element of { q ∈ Q | p ⊑ g ( q ) } Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: both f and g are monotonic g ( q ) is a greatest element of { p ∈ P | f ( p ) ≤ q } f ( p ) is a least element of { q ∈ Q | p ⊑ g ( q ) } gf is a closure operation Carl Pollard (Pre-)Algebras for Linguistics
Basic Facts about Residuated Pairs If � f, g � is a residuated pair, then: both f and g are monotonic g ( q ) is a greatest element of { p ∈ P | f ( p ) ≤ q } f ( p ) is a least element of { q ∈ Q | p ⊑ g ( q ) } gf is a closure operation fg is an interior operation Carl Pollard (Pre-)Algebras for Linguistics
Residual Operations Recall that a presemigroup is a preordered algebra � P, ⊑ , ◦� where ◦ is monotonic in both arguments and associative u.t.e. Carl Pollard (Pre-)Algebras for Linguistics
Residual Operations Recall that a presemigroup is a preordered algebra � P, ⊑ , ◦� where ◦ is monotonic in both arguments and associative u.t.e. Suppose � P, ⊑ , ◦� is a presemigroup. A binary operation ⊸ l ( ⊸ r ) on P is called a left ( right ) residual operation with respect to ◦ iff for all p, q, r ∈ P , p ◦ r ⊑ q iff r ⊑ p ⊸ l q ( r ◦ p ⊑ q iff r ⊑ p ⊸ r q ) Carl Pollard (Pre-)Algebras for Linguistics
Residual Operations Recall that a presemigroup is a preordered algebra � P, ⊑ , ◦� where ◦ is monotonic in both arguments and associative u.t.e. Suppose � P, ⊑ , ◦� is a presemigroup. A binary operation ⊸ l ( ⊸ r ) on P is called a left ( right ) residual operation with respect to ◦ iff for all p, q, r ∈ P , p ◦ r ⊑ q iff r ⊑ p ⊸ l q ( r ◦ p ⊑ q iff r ⊑ p ⊸ r q ) Left and right residual operations are antitonic in their first argument and monotonic in their second argument. Carl Pollard (Pre-)Algebras for Linguistics
Residual Operations Recall that a presemigroup is a preordered algebra � P, ⊑ , ◦� where ◦ is monotonic in both arguments and associative u.t.e. Suppose � P, ⊑ , ◦� is a presemigroup. A binary operation ⊸ l ( ⊸ r ) on P is called a left ( right ) residual operation with respect to ◦ iff for all p, q, r ∈ P , p ◦ r ⊑ q iff r ⊑ p ⊸ l q ( r ◦ p ⊑ q iff r ⊑ p ⊸ r q ) Left and right residual operations are antitonic in their first argument and monotonic in their second argument. If ◦ is commutative u.t.e., then there is no difference between a left residual operation and a right residual operation, so we speak simply of a residual operation . Carl Pollard (Pre-)Algebras for Linguistics
Residuated Presemigroups A residuated presemigroup is a tuple � P, ⊑ , ◦ , ⊸ l , ⊸ r � where � P, ⊑ , ◦� is a presemigroup with left and right residual operations ⊸ l and ⊸ r . Carl Pollard (Pre-)Algebras for Linguistics
Residuated Presemigroups A residuated presemigroup is a tuple � P, ⊑ , ◦ , ⊸ l , ⊸ r � where � P, ⊑ , ◦� is a presemigroup with left and right residual operations ⊸ l and ⊸ r . These are relevant for understanding the kind of categorial grammar called Lambek calculus . Carl Pollard (Pre-)Algebras for Linguistics
Residuated Presemigroups A residuated presemigroup is a tuple � P, ⊑ , ◦ , ⊸ l , ⊸ r � where � P, ⊑ , ◦� is a presemigroup with left and right residual operations ⊸ l and ⊸ r . These are relevant for understanding the kind of categorial grammar called Lambek calculus . Example: A ∗ with language concatenation as ◦ and the language residuals as the residual operations (see Ch. 6). Carl Pollard (Pre-)Algebras for Linguistics
Symmetric Residuated Presemigroups A symmetric residuated presemigroup is a tuple � P, ⊑ , ◦ , ⊸ � where � P, ⊑ , ◦� is a presemigroup, ◦ is commutative u.t.e., and ⊸ is a residual operation. Carl Pollard (Pre-)Algebras for Linguistics
Symmetric Residuated Presemigroups A symmetric residuated presemigroup is a tuple � P, ⊑ , ◦ , ⊸ � where � P, ⊑ , ◦� is a presemigroup, ◦ is commutative u.t.e., and ⊸ is a residual operation. These are relevant in linear logic , a kind of propositional logic that underlies certain kinds of categorial grammar, such as abstract categorial grammar and λ -grammar . Carl Pollard (Pre-)Algebras for Linguistics
Heyting Presemilattices A heyting presemilattice is a preordered algebra � P, ⊑ , ⊓ , →� where � P, ⊑ , ⊓� is a lower presemilattice, and Carl Pollard (Pre-)Algebras for Linguistics
Heyting Presemilattices A heyting presemilattice is a preordered algebra � P, ⊑ , ⊓ , →� where � P, ⊑ , ⊓� is a lower presemilattice, and → is a residual operation with respect to ⊓ . Carl Pollard (Pre-)Algebras for Linguistics
Heyting Presemilattices A heyting presemilattice is a preordered algebra � P, ⊑ , ⊓ , →� where � P, ⊑ , ⊓� is a lower presemilattice, and → is a residual operation with respect to ⊓ . The residual operation → in a heyting presemilattice is usually called a relative pseudocomplement (rpc) operation. Carl Pollard (Pre-)Algebras for Linguistics
Recommend
More recommend
