NOTATION USED IN VOLUME II

Square brackets [ ] are used for alternative readings and for reference to the

bibliography.

Let A and B be sets.—

A = : B (or B ^ A) means A is properly contained in B\

A c B (or B 2 A) means A c B or A = B\

A\B denotes the set of elements of A which are not in B;

A x B means the set of all ordered pairs (a, b) with a in A, b in B.

The signs U and n are reserved for union and intersection, respectively, of

sets and relations.

The signs v and A are used for join and meet, respectively, in [semijlattices.

| A | denotes the cardinal number of the set A.

Ko denotes the smallest infinite cardinal.

The sign o is used for composition of relations (§1.4), but is usually omitted

for composition of mappings; it is also omitted for composition of relations

in §§10.5 and 10.6.

• denotes the empty set, mapping, or relation.

t [LA] denotes the identity mapping or relation [on the set A]; see below for

convention used in §10.8.

f: A-+B means that ^ is a mapping of A into B;

f)\C means f restricted to C (C c A).

(A) denotes the subsemigroup of a semigroup 8 generated by a subset A ofS.

[A] denotes the subgroup of a group G generated by a subset A of 0\ clearly

[A] = (A U ^ "

1

) , where A^ = {a,-*: aeA}.

AB means {ab: aeA, be B}, when A and B are subsets of a semigroup S.

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[8°] means the semigroup S u 1 [S U 0] arising from a semigroup S by the

adjunction of an identity element 1 [a zero element 0], unless S already has

an identity [has a zero, and \S\ 1], in which case

S1

= S [S° = S]. (§1.1;

§6.1, p. 1.)

a p b means (a, 6) e /, where p is a relation on a set X, and a and b are elements

o f Z ;

ap denotes the set {x e X: a p x}.

S/p denotes the factor semigroup of a semigroup S modulo the congruence p

onS;

p M denotes the natural mapping a-^ap of S onto S/p. (§1.5.)

p* denotes the congruence on 8 generated by a relation p on S. (§9.2, p. 122.)

Let / be an ideal of a semigroup S.—

/ * denotes the Rees congruence t^U (/ x / ) ;

S/I denotes the Rees factor semigroup S/I*. (§1.5; §10.8, p. 227.)

Let 8 be a semigroup, and let a, beS.—

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