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.126139) from Partee (1979)
Fundamentals of Mathematics for Linguists
. The part about Lindenbaum algebras (without the
name) is subsection 5, pp. 133134.
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
(1) = 1. So far it is OK. But by the same definition
h
(
one
Mod2
+
one
Mod2
) should be equal to
h
(
one
Mod2
)
+
h
(
one
Mod2
) =
one
Mod4
+ one
Mod4
= 2. On the other hand, in
Mod2
we have
one
Mod2
+ one
Mod2
= 0 and
h
(0) = 0. So we have inconsistency
h
(
one
Mod2
+
one
Mod2
) = 2 and
h
(
one
Mod2
+
one
Mod2
) = 0, i.e.,
h
(0) = 0 and
h
(0) = 2. So there is no homomorphism from
Mod2
to
Mod4.
One of the properties of these two algebras is that the same result can be obtained in
many different ways. For example, in
Mod2
we have1 + 1 = 0 + 0 = 0. These algebras are not
free
: some nontrivial equalities hold in them. This is the reason why homomorphisms of one
Ω

algebra into another are in some cases impossible. Below we will consider
Ω
algebras which can
be homomorphically mapped to any
Ω
algebra.
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Ling 726: Mathematical Linguistics, Lec 10: Lindenbaum algebra
V. Borschev and B. Partee, October 21, 2004 p. 1
1. Freedom for algebras. Word algebras, initial algebras.
1.1. Word algebras without variables.
For any signature
Ω
there is a particularly interesting algebra called the
word algebra
which we denote as
W
Ω
.
Actually, we have considered word algebras when we have defined the
syntax of Statement Logic (and Predicate Logic). Let us do it now using the algebraic
terminology.
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 Spring '07
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 Algebra, ........., Equivalence relation, Algebraic structure, Congruence relation, Lindenbaum Algebras

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