Lecture_4_1 - Ling 726: Mathematical Linguistics, Lecture...

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Ling 726: Mathematical Linguistics, Lecture 4.1 V. Borschev and B. Partee, September 21, 2004 p. 1 Lecture 4. Algebra. Section 1: . Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras. CONTENTS 0. Why algebra?. ............................................................................................................................................................ 1 1. Algebra in a signature. ............................................................................................................................................... 1 1.0. Operations and their names. ................................................................................................................................ 1 1.1. Signature and algebra in a signature. .................................................................................................................. 2 1.2. Subalgebras. ........................................................................................................................................................ 3 2. Homomorphisms and isomorphisms; congruences and quotient algebras. ................................................................ 4 2.1. Morphisms. ......................................................................................................................................................... 4 2.2. Congruences. ...................................................................................................................................................... 6 2.3. Quotient algebras. ............................................................................................................................................... 6 Homework 4 . ................................................................................................................................................................. 7 Reading: Chapter 9: 9.1 – 9.4 of PMW, pp. 247- 253. 0. Why algebra? Really, why? Algebra is not used widely (at any rate now) in linguistics. But we think that it is very useful to know at any rate a little algebra, to be familiar with some very important basic notions like homomorphism , congruence , free algebra , etc, and with some concrete structures like lattices or Boolean algebras. Algebra gives us another point of view even on formal structures we have known before, for example, on logic. And now algebraic notions are beginning to penetrate in formal linguistic descriptions. You may have heard, for example, about unification grammars . Unification is an algebraic notion. To understand it you need to know about congruence, quotient algebras, etc. And you may have heard about the use of semi- lattice structures in Godehard Link’s and others’ work on the semantics of plurals and mass nouns. Lattice structures also seem to be relevant to OT. The first thing to realize is that “algebra” can be a count noun, not only a proper noun.
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Lecture_4_1 - Ling 726: Mathematical Linguistics, Lecture...

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