Lecture10word_alg_and_Lindenbaum_alg

Lecture10word_alg_and_Lindenbaum_alg - Ling 726:...

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Ling 726: Mathematical Linguistics, Lec 10: Lindenbaum algebra V. Borschev and B. Partee, October 21, 2004 p. 1 Lecture 10. Statement Logic as a word algebra on the set of atomic statements. Lindenbaum algebra. 0. Preliminary notes. ...................................................................................................................................................... 1 1. Freedom for algebras. Word algebras, initial algebras. ............................................................................................. 2 1.1. Word algebras without variables. ....................................................................................................................... 2 1.2. Word algebras and homomorphisms. Initial algebra. ........................................................................................ 3 2. Word algebra on a set. Free algebra. ......................................................................................................................... 5 2.1. Word algebra on a set of variables. .................................................................................................................... 5 2.2. Homomorphisms from a word algebra on a set. Free algebra. ........................................................................... 5 3. Statement Logic as a word algebra on the set of atomic statements. ......................................................................... 6 4. Lindenbaum algebra. ................................................................................................................................................. 6 Homework 11. ............................................................................................................................................................... 7 Reading : Previously distributed extract, “Boolean algebras” (pp.126-139) from Partee (1979) Fundamentals of Mathematics for Linguists . The part about Lindenbaum algebras (without the name) is subsection 5, pp. 133-134. The algebraic parts of the handout are based on the PtMW textbook and two other sources: Cohn P.M. Universal algebra, Harper and Row. New York, Evanston and London, 1965. Burstall R.M. and J.A. Goguen, Algebras, Theories and Freeness: An Introduction for Computer Scientists. In: M. Broy and G. Schmidt (eds.) Theoretical Foundations of Programming Methodology, Reidel, 1982, pp. 329 – 349. Note: Lecture 10 in 2004 corresponds to lectures 8 and 9 in 2001. Homework 11 in 2004 corresponds to homework 9 in 2001. [We are omitting the earlier homework 8, on congruences.] 0. Preliminary notes. Let us return to the algebras considered in Lecture 4 (7). We considered the homomorphism f : Mod4 Mod2 . Are there any homomorphisms from Mod2 to Mod4 ? We show that there are none. Suppose that h : Mod2 Mod4 is such a homomorphism. Then, by definition of a homomorphism, h ( zero Mod2 ) = zero Mod4 , i.e. h (0) = 0, h ( one Mod2 ) = one Mod4 , i.e. h
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This note was uploaded on 02/22/2012 for the course LINGUIST 726 taught by Professor Partee during the Spring '07 term at UMass (Amherst).

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Lecture10word_alg_and_Lindenbaum_alg - Ling 726:...

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