(Pre-)Algebras for Linguistics
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1 1. Review of Preorders Linguistics 680: Formal Foundations Autumn 2010
2 (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation ( less than or equivalent to ) on A which is reflexive and transitive.
3 (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation ( less than or equivalent to ) on A which is reflexive and transitive. An antisymmetric preorder is called an order.
4 (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation ( 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 b and b a.
5 (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation ( 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 b and b a. If is an order, then is just the identity relation on A, and correspondingly is read as less than or equal to.
6 Background Assumptions (until further notice) is a preorder on A
7 Background Assumptions (until further notice) is a preorder on A is the induced equivalence relation
8 Background Assumptions (until further notice) is a preorder on A is the induced equivalence relation S A
9 Background Assumptions (until further notice) is a preorder on A is the induced equivalence relation S A a A (not necessarily S)
10 More Definitions We call a an upper (lower) bound of S iff, for every b S, b a (a b).
11 More Definitions We call a an upper (lower) bound of S iff, for every b S, b a (a 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
12 More Definitions We call a an upper (lower) bound of S iff, for every b S, b a (a 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
13 More Definitions We call a an upper (lower) bound of S iff, for every b S, b a (a 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 b (b a), then a b.
14 More Definitions We call a an upper (lower) bound of S iff, for every b S, b a (a 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 b (b a), then a b. Note: the definition of greatest/least above is equivalent to the one in Chapter 3.
15 Observations If a is greatest (least) in S, it is maximal (minimal) in S.
16 Observations If a is greatest (least) in S, it is maximal (minimal) in S. All greatest (least) members of S are equivalent.
17 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.
18 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 is an order, S has at most one greatest (least) member, and A has at most one top (bottom).
19 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.
20 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.
21 More about LUBs and GLBs Any greatest (least) member of S is a lub (glb) of S.
22 More about LUBs and GLBs Any greatest (least) member of S is a lub (glb) of S. All lubs (glbs) of S are equivalent.
23 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 is an order, then S has at most one lub (glb).
24 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 is an order, then S has at most one lub (glb). A lub (glb) of A is the same thing as a top (bottom).
25 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 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).
26 Some Notation If S = {a}, then UB(S) (LB(S)) is usually written a ( a), read up of a ( down of a ).
27 Some Notation If S = {a}, then UB(S) (LB(S)) is usually written a ( a), read up of a ( down of a ). If S has a unique glb (lub), it is written S ( S).
28 Some Notation If S = {a}, then UB(S) (LB(S)) is usually written a ( a), read up of a ( down of a ). If S has a unique glb (lub), it is written S ( S). If S = {a, b} and S has a unique glb (lub), it is written a b (a b).
29 Some Notation If S = {a}, then UB(S) (LB(S)) is usually written a ( a), read up of a ( down of a ). If S has a unique glb (lub), it is written S ( S). If S = {a, b} and S has a unique glb (lub), it is written a b (a b). If A has a unique top (bottom), it is written ( ).
30 Facts about and when is an order Idempotence a a exists and equals a.
31 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal.
32 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal. Associativity If (a b) c and a (b c) both exist, they are equal.
33 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal. Associativity If (a b) c and a (b c) both exist, they are equal. The preceding three assertions remain true if is replaced by.
34 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal. Associativity If (a b) c and a (b c) both exist, they are equal. The preceding three assertions remain true if is replaced by. Interdefinability a b iff a b exists and equals a iff a b exists and equals b.
35 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal. Associativity If (a b) c and a (b c) both exist, they are equal. The preceding three assertions remain true if is replaced by. Interdefinability a b iff a b exists and equals a iff a b exists and equals b. Absorbtion If (a b) b exists, it equals b.
36 Facts about and when is an order Idempotence a a exists and equals a. Commutativity If a b exists, so does b a, and they are equal. Associativity If (a b) c and a (b c) both exist, they are equal. The preceding three assertions remain true if is replaced by. Interdefinability a b iff a b exists and equals a iff a b exists and equals b. Absorbtion If (a b) b exists, it equals b. If (a b) b exists, it equals b.
37 Monotonicity, Antitonicity, and Tonicity Suppose A and B are preordered by and respectively. Then a function f : A B is called: monotonic or order-preserving iff, for all a, a A, if a a, then f(a) f(a );
38 Monotonicity, Antitonicity, and Tonicity Suppose A and B are preordered by and respectively. Then a function f : A B is called: monotonic or order-preserving iff, for all a, a A, if a a, then f(a) f(a ); antitonic or order-reversing iff, for all a, a A, if a a, then f(a ) f(a); and
39 Monotonicity, Antitonicity, and Tonicity Suppose A and B are preordered by and respectively. Then a function f : A B is called: monotonic or order-preserving iff, for all a, a A, if a a, then f(a) f(a ); antitonic or order-reversing iff, for all a, a A, if a a, then f(a ) f(a); and tonic iff it is either monotonic or antitonic.
40 Preorder (Anti-)Isomorphism A monotonic (antitonic) bijection is called a preorder isomorphism (preorder anti-isomorphism) provided its inverse is also monotonic (antitonic).
41 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.
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