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.
We will look at a few kinds of algebras and at some fundamental notions used in algebra in
general.
1. Algebra in a signature.
1.0. Operations and their names.
Roughly speaking, an algebra is a set and a collection of operations on this set. For example, the
set of natural numbers with the operations of addition and multiplication form an algebra. But we
would like to consider different sets with “the same” or similar operations on them, for example,
addition and multiplication on the set of rational numbers, or on other sets. So we need names for
operations. A set of names of operations is called a
signature
.
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Ling 726: Mathematical Linguistics, Lecture 4.1
V. Borschev and B. Partee, September 21, 2004
p. 2
1.1. Signature and algebra in a signature.
Let
N
= {0,1,2,…}
be as usual the set of natural numbers.
[Review Partee 1979, Chapter 0]
Definition
. A set
Ω
(of symbols) together with a function
a
:
Ω
→
N
is called a
signature
. Its
elements are called names (symbols) of operations or
operators
. If
a
(
ω
) =
n
, we say that operator
ω
is
n
ary. We write
Ω
(
n
) = {
ω
∈
Ω
a
(
ω
) =
n
}. [Note that in this case the notation
Ω
(
n
) does
not
represent functionargument application. This use, where the
n
is ‘indexing’ a particular
subset of
Ω
, is also fairly common.]
Note that for
n
= 0, an
n
ary
operator becomes 0ary.
n
ary operators will name
n
ary
operations of the kind
f
:
A
n
→
A
on some set
A
. In the case
n
= 0 we have 0ary operations on
the set
A
. Every
0ary operation just marks some element in
A
.
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