(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 2 A (not necessarily 2 S) Background Assumptions (until further notice) v is a preorder on A Carl Pollard (Pre-)Algebras for Linguistics S ⊆ A a 2 A (not necessarily 2 S) Background Assumptions (until further notice) v is a preorder on A ≡ is the induced equivalence relation Carl Pollard (Pre-)Algebras for Linguistics a 2 A (not necessarily 2 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 2 A (not necessarily 2 S) Carl Pollard (Pre-)Algebras for Linguistics Suppose moreover that a 2 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 2 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 2 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 2 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 2 S, b v a (a v b). Suppose moreover that a 2 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 2 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 2 S, b v a (a v b). Suppose moreover that a 2 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 2 S, b v a (a v b). Suppose moreover that a 2 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 2 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 2 S, b v a (a v b). Suppose moreover that a 2 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 2 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.