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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