(Pre-)Algebras for Linguistics
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1 2. Introducing Preordered Algebras Linguistics 680: Formal Foundations Autumn 2010
2 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations).
3 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations). Some of the simplest algebras are ones with just a single binary operation. Some important examples: Groupoids: no restrictions on.
4 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations). Some of the simplest algebras are ones with just a single binary operation. Some important examples: Groupoids: no restrictions on. Semigroups: is associative.
5 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations). Some of the simplest algebras are ones with just a single binary operation. Some important examples: Groupoids: no restrictions on. Semigroups: is associative. Commutative semigroups: is associative and commutative.
6 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations). Some of the simplest algebras are ones with just a single binary operation. Some important examples: Groupoids: no restrictions on. Semigroups: is associative. Commutative semigroups: is associative and commutative. Semilattices: is associative, commutative, and idempotent.
7 Algebras A (one-sorted) algebra is a set with one or more operations (where special elements are thought of as nullary operations). Some of the simplest algebras are ones with just a single binary operation. Some important examples: Groupoids: no restrictions on. Semigroups: is associative. Commutative semigroups: is associative and commutative. Semilattices: is associative, commutative, and idempotent. A monoid is a semigroup with a two-sided identity element e.
8 Tonicity Generalized Recall: a unary operation on a (pre)order is called tonic provided it is either monotonic or antitonic.
9 Tonicity Generalized Recall: a unary operation on a (pre)order is called tonic provided it is either monotonic or antitonic. An operation of arbitrary arity on a (pre)order is called tonic if it is tonic in each argument as the other arguments are held fixed. All nullary operations are (trivially) tonic.
10 Tonicity Generalized Recall: a unary operation on a (pre)order is called tonic provided it is either monotonic or antitonic. An operation of arbitrary arity on a (pre)order is called tonic if it is tonic in each argument as the other arguments are held fixed. All nullary operations are (trivially) tonic. The two definitions coincide in the unary case.
11 Tonicity Generalized Recall: a unary operation on a (pre)order is called tonic provided it is either monotonic or antitonic. An operation of arbitrary arity on a (pre)order is called tonic if it is tonic in each argument as the other arguments are held fixed. All nullary operations are (trivially) tonic. The two definitions coincide in the unary case. a binary operation is tonic iff (1) for each a, the function that maps each b to a b is tonic, and (2) for each b, the function that maps each a to a b is tonic.
12 (Pre)ordered Algebras A (pre)ordered algebra is a (pre)order which is also an algebra whose operations are all tonic.
13 (Pre)ordered Algebras A (pre)ordered algebra is a (pre)order which is also an algebra whose operations are all tonic. An operation in a preordered algebra is said to have a property up to equivalence (u.t.e.) if it holds with = replaced by, where is the equivalence relation induced by the preorder.
14 (Pre)ordered Algebras A (pre)ordered algebra is a (pre)order which is also an algebra whose operations are all tonic. An operation in a preordered algebra is said to have a property up to equivalence (u.t.e.) if it holds with = replaced by, where is the equivalence relation induced by the preorder. For example, is commutative u.t.e. iff for all a and b, a b b a.
15 Some Kinds of Preordered Algebras For future reference: A pregroupoid is a preorder with one binary operation which is monotonic on both arguments.
16 Some Kinds of Preordered Algebras For future reference: A pregroupoid is a preorder with one binary operation which is monotonic on both arguments. A presemigroup is a pregroupoid whose operation is associative u.t.e.
17 Some Kinds of Preordered Algebras For future reference: A pregroupoid is a preorder with one binary operation which is monotonic on both arguments. A presemigroup is a pregroupoid whose operation is associative u.t.e. A presemilattice is a presemigroup which is both commutative u.t.e. and idempotent u.t.e.
18 Some Kinds of Preordered Algebras For future reference: A pregroupoid is a preorder with one binary operation which is monotonic on both arguments. A presemigroup is a pregroupoid whose operation is associative u.t.e. A presemilattice is a presemigroup which is both commutative u.t.e. and idempotent u.t.e. A premonoid is a presemigroup with an additional unary operation e which is a two-sided identity u.t.e.
19 Some Kinds of Preordered Algebras For future reference: A pregroupoid is a preorder with one binary operation which is monotonic on both arguments. A presemigroup is a pregroupoid whose operation is associative u.t.e. A presemilattice is a presemigroup which is both commutative u.t.e. and idempotent u.t.e. A premonoid is a presemigroup with an additional unary operation e which is a two-sided identity u.t.e. Note: A prewidget is a widget if it is antisymmetric, but not otherwise!
20 Substitutivity u.t.e Preordered algebras enjoy the property of substitutivity u.t.e, i.e. replacing the arguments of any operation by equivalents yields an equivalent result.
21 Substitutivity u.t.e Preordered algebras enjoy the property of substitutivity u.t.e, i.e. replacing the arguments of any operation by equivalents yields an equivalent result. For example, in the binary case, this means that if a b and c d, then a c b d.
22 Some Kinds of Ordered Algebras For future reference: An ordered groupoid is an antisymmetric pregroupoid.
23 Some Kinds of Ordered Algebras For future reference: An ordered groupoid is an antisymmetric pregroupoid. An ordered semigroup is an antisymmetric presemigroup.
24 Some Kinds of Ordered Algebras For future reference: An ordered groupoid is an antisymmetric pregroupoid. An ordered semigroup is an antisymmetric presemigroup. An ordered semilattice is an antisymmetric presemilattice.
25 Some Kinds of Ordered Algebras For future reference: An ordered groupoid is an antisymmetric pregroupoid. An ordered semigroup is an antisymmetric presemigroup. An ordered semilattice is an antisymmetric presemilattice. An ordered monoid is an antisymmetric premonoid.
26 An Important Example of an Ordered Monoid For any set A, we have seen that (A ) forms a monoid with A-languages as the elements
27 An Important Example of an Ordered Monoid For any set A, we have seen that (A ) forms a monoid with A-languages as the elements (language concatenation) as the binary operation
28 An Important Example of an Ordered Monoid For any set A, we have seen that (A ) forms a monoid with A-languages as the elements (language concatenation) as the binary operation 1 A as the two-sided identity.
29 An Important Example of an Ordered Monoid For any set A, we have seen that (A ) forms a monoid with A-languages as the elements (language concatenation) as the binary operation 1 A as the two-sided identity. We turn this into an ordered monoid by taking the order to be subset inclusion of languages. (You need to check that is monotonic in both arguments.)
30 Two Important Examples of an Ordered Semilattice In both examples, we take the order to be the subset inclusion ordering on (A), for some set A. Example 1: take the binary operation to be set intersection. Observation: a b iff a b = a.
31 Two Important Examples of an Ordered Semilattice In both examples, we take the order to be the subset inclusion ordering on (A), for some set A. Example 1: take the binary operation to be set intersection. Observation: a b iff a b = a. Example 2: take the binary operation to be set union. Observation: a b iff a b = b. These observations motivate the following definitions.
32 Two Kinds of Presemilattices Suppose P,, is a presemilattice, i.e. is monotonic in both arguments, associative u.t.e., commutative u.t.e., and idempotent u.t.e. Then it is called: upper iff, for all a, b P, a b iff a b b.
33 Two Kinds of Presemilattices Suppose P,, is a presemilattice, i.e. is monotonic in both arguments, associative u.t.e., commutative u.t.e., and idempotent u.t.e. Then it is called: upper iff, for all a, b P, a b iff a b b. lower iff, for all a, b P, a b iff a b a.
34 A Theorem about Presemilattices In an upper presemilattice, is a lub operation (hence usually written ).
35 A Theorem about Presemilattices In an upper presemilattice, is a lub operation (hence usually written ). In a lower presemilattice, is a glb operation. (hence usually written ).
36 A Theorem about lubs and glbs Suppose P,, is a preorder with a (not necessarily tonic) binary operation. Then if is a lub operation, then in fact P,, is an upper presemilattice.
37 A Theorem about lubs and glbs Suppose P,, is a preorder with a (not necessarily tonic) binary operation. Then if is a lub operation, then in fact P,, is an upper presemilattice. if is a glb operation, then in fact P,, is a lower presemilattice.
38 The Semantics of and and or We can use presemilattices to analyze the meanings of the two English words and and or (in-class exercise).
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