(Pre-)Algebras for Linguistics 1. Review of Preorders

Carl Pollard

Linguistics 680: Formal Foundations

Autumn 2010

Carl Pollard (Pre-)Algebras for Linguistics An antisymmetric preorder is called an order. The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as ‘less than or equal to’.

(Pre-)Orders and Induced Equivalence

A preorder on a set A is a binary relation v (‘less than or equivalent to’) on A which is reflexive and transitive.

Carl Pollard (Pre-)Algebras for Linguistics The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as ‘less than or equal to’.

(Pre-)Orders and Induced Equivalence

A preorder on a set A is a binary relation v (‘less than or equivalent to’) on A which is reflexive and transitive. An antisymmetric preorder is called an order.

Carl Pollard (Pre-)Algebras for Linguistics If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as ‘less than or equal to’.

(Pre-)Orders and Induced Equivalence

A preorder on a set A is a binary relation v (‘less than or equivalent to’) on A which is reflexive and transitive. An antisymmetric preorder is called an order. The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a.

Carl Pollard (Pre-)Algebras for Linguistics (Pre-)Orders and Induced Equivalence

A preorder on a set A is a binary relation v (‘less than or equivalent to’) on A which is reflexive and transitive. An antisymmetric preorder is called an order. The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as ‘less than or equal to’.

Carl Pollard (Pre-)Algebras for Linguistics ≡ is the induced equivalence relation S ⊆ A a ∈ A (not necessarily ∈ S)

Background Assumptions (until further notice)

v is a preorder on A

Carl Pollard (Pre-)Algebras for Linguistics S ⊆ A a ∈ A (not necessarily ∈ S)

Background Assumptions (until further notice)

v is a preorder on A ≡ is the induced equivalence relation

Carl Pollard (Pre-)Algebras for Linguistics a ∈ A (not necessarily ∈ S)

Background Assumptions (until further notice)

v is a preorder on A ≡ is the induced equivalence relation S ⊆ A

Carl Pollard (Pre-)Algebras for Linguistics Background Assumptions (until further notice)

v is a preorder on A ≡ is the induced equivalence relation S ⊆ A a ∈ A (not necessarily ∈ S)

Carl Pollard (Pre-)Algebras for Linguistics Suppose moreover that a ∈ S. Then a is said to be: greatest (least) in S iff it is an upper (lower) bound of S a top (bottom) iff it is greatest (least) in A maximal (minimal) in S iff, for every b ∈ S, if a v b (b v a), then a ≡ b. Note: the definition of greatest/least above is equivalent to the one in Chapter 3.

More Definitions

We call a an upper (lower) bound of S iff, for every b ∈ S, b v a (a v b).

Carl Pollard (Pre-)Algebras for Linguistics a top (bottom) iff it is greatest (least) in A maximal (minimal) in S iff, for every b ∈ S, if a v b (b v a), then a ≡ b. Note: the definition of greatest/least above is equivalent to the one in Chapter 3.

More Definitions

We call a an upper (lower) bound of S iff, for every b ∈ S, b v a (a v b). Suppose moreover that a ∈ S. Then a is said to be: greatest (least) in S iff it is an upper (lower) bound of S

Carl Pollard (Pre-)Algebras for Linguistics maximal (minimal) in S iff, for every b ∈ S, if a v b (b v a), then a ≡ b. Note: the definition of greatest/least above is equivalent to the one in Chapter 3.

More Definitions

We call a an upper (lower) bound of S iff, for every b ∈ S, b v a (a v b). Suppose moreover that a ∈ S. Then a is said to be: greatest (least) in S iff it is an upper (lower) bound of S a top (bottom) iff it is greatest (least) in A

Carl Pollard (Pre-)Algebras for Linguistics Note: the definition of greatest/least above is equivalent to the one in Chapter 3.

More Definitions

We call a an upper (lower) bound of S iff, for every b ∈ S, b v a (a v b). Suppose moreover that a ∈ S. Then a is said to be: greatest (least) in S iff it is an upper (lower) bound of S a top (bottom) iff it is greatest (least) in A maximal (minimal) in S iff, for every b ∈ S, if a v b (b v a), then a ≡ b.

Carl Pollard (Pre-)Algebras for Linguistics More Definitions

We call a an upper (lower) bound of S iff, for every b ∈ S, b v a (a v b). Suppose moreover that a ∈ S. Then a is said to be: greatest (least) in S iff it is an upper (lower) bound of S a top (bottom) iff it is greatest (least) in A maximal (minimal) in S iff, for every b ∈ S, if a v b (b v a), then a ≡ b. Note: the definition of greatest/least above is equivalent to the one in Chapter 3.

Carl Pollard (Pre-)Algebras for Linguistics All greatest (least) members of S are equivalent. And so all tops (bottoms) of A are equivalent. And so if v is an order, S has at most one greatest (least) member, and A has at most one top (bottom).

Observations

If a is greatest (least) in S, it is maximal (minimal) in S.

Carl Pollard (Pre-)Algebras for Linguistics And so all tops (bottoms) of A are equivalent. And so if v is an order, S has at most one greatest (least) member, and A has at most one top (bottom).

Observations

If a is greatest (least) in S, it is maximal (minimal) in S. All greatest (least) members of S are equivalent.

Carl Pollard (Pre-)Algebras for Linguistics And so if v is an order, S has at most one greatest (least) member, and A has at most one top (bottom).

Observations

If a is greatest (least) in S, it is maximal (minimal) in S. All greatest (least) members of S are equivalent. And so all tops (bottoms) of A are equivalent.

Carl Pollard (Pre-)Algebras for Linguistics Observations

If a is greatest (least) in S, it is maximal (minimal) in S. All greatest (least) members of S are equivalent. And so all tops (bottoms) of A are equivalent. And so if v is an order, S has at most one greatest (least) member, and A has at most one top (bottom).

Carl Pollard (Pre-)Algebras for Linguistics A greatest member of LB(S) is called a greatest lower bound (glb) of S.

LUBs and GLBs

Let UB(S) (LB(S)) be the set of upper (lower) bounds of S. A least member of UB(S) is called a least upper bound (lub) of S.

Carl Pollard (Pre-)Algebras for Linguistics LUBs and GLBs

Let UB(S) (LB(S)) be the set of upper (lower) bounds of S. A least member of UB(S) is called a least upper bound (lub) of S. A greatest member of LB(S) is called a greatest lower bound (glb) of S.

Carl Pollard (Pre-)Algebras for Linguistics All lubs (glbs) of S are equivalent. If v is an order, then S has at most one lub (glb). A lub (glb) of A is the same thing as a top (bottom). A lub (glb) of ∅ is the same thing as a bottom (top).

More about LUBs and GLBs

Any greatest (least) member of S is a lub (glb) of S.

Carl Pollard (Pre-)Algebras for Linguistics If v is an order, then S has at most one lub (glb). A lub (glb) of A is the same thing as a top (bottom). A lub (glb) of ∅ is the same thing as a bottom (top).

More about LUBs and GLBs

Any greatest (least) member of S is a lub (glb) of S. All lubs (glbs) of S are equivalent.

Carl Pollard (Pre-)Algebras for Linguistics A lub (glb) of A is the same thing as a top (bottom). A lub (glb) of ∅ is the same thing as a bottom (top).

More about LUBs and GLBs

Any greatest (least) member of S is a lub (glb) of S. All lubs (glbs) of S are equivalent. If v is an order, then S has at most one lub (glb).

Carl Pollard (Pre-)Algebras for Linguistics A lub (glb) of ∅ is the same thing as a bottom (top).

More about LUBs and GLBs

Any greatest (least) member of S is a lub (glb) of S. All lubs (glbs) of S are equivalent. If v is an order, then S has at most one lub (glb). A lub (glb) of A is the same thing as a top (bottom).

Carl Pollard (Pre-)Algebras for Linguistics More about LUBs and GLBs

Any greatest (least) member of S is a lub (glb) of S. All lubs (glbs) of S are equivalent. If v is an order, then S has at most one lub (glb). A lub (glb) of A is the same thing as a top (bottom). A lub (glb) of ∅ is the same thing as a bottom (top).

Carl Pollard (Pre-)Algebras for Linguistics F If S has a unique glb (lub), it is written d S ( S). If S = {a, b} and S has a unique glb (lub), it is written a u b (a t b). If A has a unique top (bottom), it is written > (⊥).

Some Notation

If S = {a}, then UB(S) (LB(S)) is usually written ↑ a (↓ a), read ‘up of a’ (‘down of a’).

Carl Pollard (Pre-)Algebras for Linguistics If S = {a, b} and S has a unique glb (lub), it is written a u b (a t b). If A has a unique top (bottom), it is written > (⊥).

Some Notation

If S = {a}, then UB(S) (LB(S)) is usually written ↑ a (↓ a), read ‘up of a’ (‘down of a’). F If S has a unique glb (lub), it is written d S ( S).

Carl Pollard (Pre-)Algebras for Linguistics If A has a unique top (bottom), it is written > (⊥).

Some Notation

If S = {a}, then UB(S) (LB(S)) is usually written ↑ a (↓ a), read ‘up of a’ (‘down of a’). F If S has a unique glb (lub), it is written d S ( S). If S = {a, b} and S has a unique glb (lub), it is written a u b (a t b).

Carl Pollard (Pre-)Algebras for Linguistics Some Notation

If S = {a}, then UB(S) (LB(S)) is usually written ↑ a (↓ a), read ‘up of a’ (‘down of a’). F If S has a unique glb (lub), it is written d S ( S). If S = {a, b} and S has a unique glb (lub), it is written a u b (a t b). If A has a unique top (bottom), it is written > (⊥).

Carl Pollard (Pre-)Algebras for Linguistics Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a.

Carl Pollard (Pre-)Algebras for Linguistics Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal.

Carl Pollard (Pre-)Algebras for Linguistics The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal.

Carl Pollard (Pre-)Algebras for Linguistics Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t.

Carl Pollard (Pre-)Algebras for Linguistics Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b.

Carl Pollard (Pre-)Algebras for Linguistics If (a t b) u b exists, it equals b.

Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b.

Carl Pollard (Pre-)Algebras for Linguistics Facts about u and t when v is an order

Idempotence a u a exists and equals a. Commutativity If a u b exists, so does b u a, and they are equal. Associativity If (a u b) u c and a u (b u c) both exist, they are equal. The preceding three assertions remain true if u is replaced by t. Interdefinability a v b iff a u b exists and equals a iff a t b exists and equals b. Absorbtion If (a u b) t b exists, it equals b. If (a t b) u b exists, it equals b.

Carl Pollard (Pre-)Algebras for Linguistics antitonic or order-reversing iff, for all a, a0 ∈ A, if a v a0, then f(a0) ≤ f(a); and tonic iff it is either monotonic or antitonic.

Monotonicity, Antitonicity, and Tonicity

Suppose A and B are preordered by v and ≤ respectively. Then a function f : A → B is called: monotonic or order-preserving iff, for all a, a0 ∈ A, if a v a0, then f(a) ≤ f(a0);

Carl Pollard (Pre-)Algebras for Linguistics tonic iff it is either monotonic or antitonic.

Monotonicity, Antitonicity, and Tonicity

Suppose A and B are preordered by v and ≤ respectively. Then a function f : A → B is called: monotonic or order-preserving iff, for all a, a0 ∈ A, if a v a0, then f(a) ≤ f(a0); antitonic or order-reversing iff, for all a, a0 ∈ A, if a v a0, then f(a0) ≤ f(a); and

Carl Pollard (Pre-)Algebras for Linguistics Monotonicity, Antitonicity, and Tonicity

Suppose A and B are preordered by v and ≤ respectively. Then a function f : A → B is called: monotonic or order-preserving iff, for all a, a0 ∈ A, if a v a0, then f(a) ≤ f(a0); antitonic or order-reversing iff, for all a, a0 ∈ A, if a v a0, then f(a0) ≤ f(a); and tonic iff it is either monotonic or antitonic.

Carl Pollard (Pre-)Algebras for Linguistics Two preordered sets are said to be preorder-isomorphic provided there is a preorder isomorphism from one to the other.

Preorder (Anti-)Isomorphism

A monotonic (antitonic) bijection is called a preorder isomorphism (preorder anti-isomorphism) provided its inverse is also monotonic (antitonic).

Carl Pollard (Pre-)Algebras for Linguistics Preorder (Anti-)Isomorphism

A monotonic (antitonic) bijection is called a preorder isomorphism (preorder anti-isomorphism) provided its inverse is also monotonic (antitonic). Two preordered sets are said to be preorder-isomorphic provided there is a preorder isomorphism from one to the other.

Carl Pollard (Pre-)Algebras for Linguistics