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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