NOTATION USED IN VOLUME ONE
Square brackets are used for alternative readings and for reference to the
bibliography.
Let A and B be sets.
A C B (or Bo A) means A is properly contained in B.
A c B (or B 2 A) means A C B or A = B.
A\B means the set of elements of A which are not in J5.
Ax B means the set of all ordered pairs (a, b) with a in A, b in J3.
The signs U and n are reserved for union and intersection, respectively, of
sets and relations. The signs v and A will be used for join and meet in
[semi]lattices.
|-41 means the cardinal number of the set A.
The sign o is used for composition of relations (§1.4), but is usually omitted
for composition of mappings.
denotes the empty set, mapping, or relation.
i [iA] denotes the identity mapping or relation [on the set A],
If ^ is a mapping whose domain includes A, then f\A means f restricted
to A.
{#!,• •, an} means the set whose members are ai,« •, an. Braces are
sometimes omitted on single elements, for example Aub instead of A U {&}.
If P(x) is a proposition for each element a: of a set X, then the set of all
a; in I for which P(x) is true is denoted by either {xeX: P(x)} or
{x:P(x),xeX}.
If M(x) is a set for each a; in a set X, then the union of all the sets M(x)
with x in X is denoted by either \JXexM(x) or \J{M(x)
:XEX}.
If F(x) is a member of a set C for each x in a set X, then the subset of C
consisting of all F(x) with x in X is denoted by {F(x): x e X). If X = A x B,
we may write {F(a, b):aGA,beB} instead of {F(a, b): (a, b)eAx B}.
If A is a subset of a semigroup 8, then ^4 denotes the subsemigroup of 8
generated by A. If S is a group, then the subgroup of 8 generated by A is
( i u i
- 1
) , where
A~l
= { a ^ i a e i } .
If ^4 and B are subsets of a semigroup 8, then ^4J5 means {ab:aeA,
beB}.
S1
[S°] means the semigroup /Sul[SuO ] arising from a semigroup S by
the adjunction of an identity element 1 [a zero element 0], unless 8 already
has an identity [has a zero, and \S\ 1], in which case
S1
= 8 [S° = 8], (§1.1)
a\b means "a divides b", that is,
beaSl
where a and b are elements of a
commutative semigroup 8. (§4.3)
pa [VI denotes the inner right [left] translation x-xa[x-ax] of a
semigroup S, where a is a fixed element of 8. (§1.3)
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