Conventions on
Terminology
We assume that our readers are acquainted with the notion of a set (and even with
groups and rings, though their definitions will be recalled in Chapters 2, 4). They will
have seen notations such as x E 5 (x is a member of 5), 5' S; 5 or 5 ;2 5' (5' is a
subset of 5) and T C 5 or 5 :) T (T is a proper subset of 5) and 0 for the empty
set. For any propositions P, Q we write 'p::::} Q' or 'Q {::: P' to indicate that P
implies Q, and 'p # Q' to mean 'p::::} Q and Q::::} P', i.e. that P is equivalent to Q.
A property (of members of a set 5) is said to hold for almost all members of 5 if it
holds for all but a finite number of members of 5. If T is a subset of 5, its comple-
ment in 5 will be denoted by 5\ T. This notation is also used occasionally for the left
coset space (see Section 2.1); the risk of confusion is small.
We can list the elements of a set 5 by indexing them, e.g. if 5 is finite, with n
elements, we can write 5 = {Xl, X2, ... , x n }; we also write 151 = n. More generally,
any set can be indexed by a suitable indexing set: 5 = {X).hE!' where I is the indexing
set. A set in this form is often called a family indexed by I; it is in effect prescribing a
mapping from I to 5. This mapping is generally not assumed to be injective, thus x).
may equal xfJ- even if A =/: M.
All mappings between sets are as a rule written on the right, so that fg means: first
f, then g. Iff: 5 ---+ T, i.e. f is a mapping from 5 to T and 5' is a subset of 5, then the
restriction of f to 5' is denoted by fI5'. A mapping f : 5 ---+ T is called injective or
one-one if different members of 5 have different images, surjective or onto if every
member of T is an image of some member of 5, and bijective if it is both injective
and surjective. Mappings are often arranged as diagrams (see Section 4.2); a diagram
is commutative if the different ways of going from one point to another along the
arrows give the same result.
Frequently a two-index expression f(i, j) is equal to 1 if i = j and 0 otherwise. This
is indicated by using the Kronecker symbol 8ij ; thus f(i, j) = 8ij •
A set 5 is partially ordered, often just called ordered, if there is a binary relation :s ,
called a partial ordering, defined on 5 with the properties:
0.1 x:s X for all X E 5 (reflexive),
0.2 X :s y, y :s z ::::} x :s z for all x, y, z E 5 (transitive),
0.3 x :s y, y :s x ::::} x = y for all x, y E 5 (antisymmetric).
If only 0.1 and 0.2 hold, we speak of a preordering.
The ordering is total if any two elements are comparable, i.e. x :s y or y :s x for
any x, y E 5. If ':s' is a partial ordering on a set 5, we shall write 'x < y' (x is strictly
xi
Terminology
We assume that our readers are acquainted with the notion of a set (and even with
groups and rings, though their definitions will be recalled in Chapters 2, 4). They will
have seen notations such as x E 5 (x is a member of 5), 5' S; 5 or 5 ;2 5' (5' is a
subset of 5) and T C 5 or 5 :) T (T is a proper subset of 5) and 0 for the empty
set. For any propositions P, Q we write 'p::::} Q' or 'Q {::: P' to indicate that P
implies Q, and 'p # Q' to mean 'p::::} Q and Q::::} P', i.e. that P is equivalent to Q.
A property (of members of a set 5) is said to hold for almost all members of 5 if it
holds for all but a finite number of members of 5. If T is a subset of 5, its comple-
ment in 5 will be denoted by 5\ T. This notation is also used occasionally for the left
coset space (see Section 2.1); the risk of confusion is small.
We can list the elements of a set 5 by indexing them, e.g. if 5 is finite, with n
elements, we can write 5 = {Xl, X2, ... , x n }; we also write 151 = n. More generally,
any set can be indexed by a suitable indexing set: 5 = {X).hE!' where I is the indexing
set. A set in this form is often called a family indexed by I; it is in effect prescribing a
mapping from I to 5. This mapping is generally not assumed to be injective, thus x).
may equal xfJ- even if A =/: M.
All mappings between sets are as a rule written on the right, so that fg means: first
f, then g. Iff: 5 ---+ T, i.e. f is a mapping from 5 to T and 5' is a subset of 5, then the
restriction of f to 5' is denoted by fI5'. A mapping f : 5 ---+ T is called injective or
one-one if different members of 5 have different images, surjective or onto if every
member of T is an image of some member of 5, and bijective if it is both injective
and surjective. Mappings are often arranged as diagrams (see Section 4.2); a diagram
is commutative if the different ways of going from one point to another along the
arrows give the same result.
Frequently a two-index expression f(i, j) is equal to 1 if i = j and 0 otherwise. This
is indicated by using the Kronecker symbol 8ij ; thus f(i, j) = 8ij •
A set 5 is partially ordered, often just called ordered, if there is a binary relation :s ,
called a partial ordering, defined on 5 with the properties:
0.1 x:s X for all X E 5 (reflexive),
0.2 X :s y, y :s z ::::} x :s z for all x, y, z E 5 (transitive),
0.3 x :s y, y :s x ::::} x = y for all x, y E 5 (antisymmetric).
If only 0.1 and 0.2 hold, we speak of a preordering.
The ordering is total if any two elements are comparable, i.e. x :s y or y :s x for
any x, y E 5. If ':s' is a partial ordering on a set 5, we shall write 'x < y' (x is strictly
xi
