a1 denotes the congruence t U [a n (Dn x Z)w)] \J (In x In) on ^ y determined
by the congruence a on In+i/In, with w a positive integer. (§10.8, p. 228.)
Let a, /? be elements of &~x, and £ an infinite cardinal.—
Xo = Xo (a, j8) denotes {x e X : xa ^ xjS};
dr(a, /?), the difference rank of the pair (a, /?), is zero if a = /J and max{|Xo«|,
|Xoj8|} otherwise;
A^ denotes the congruence {(a, /?) E ZTX X $~X : dr(a, /?) £} on ^ x infinite or
| = 1 ) . (§10.8, p. 228.)
rj(p) means the primary cardinal of the congruence p on 3~x; ^(0 = 1, and, if
p ^ t , -Mp)
^he p-class containing transformations of rank 1. (§10.8,
Let A be a cardinal satisfying -q(p) ^ \X\.—
A* means the smallest cardinal exceeding every cardinal in the set {dr(a, j8):
(a, P)ep, rank a = rank p = A}. (§10.8, p. 234.)
Ms [sM] denotes a right [left] operand over a semigroup S. (§11.1, p. 250.)
SMT denotes a bioperand over the semigroups S and T. (§11.1, p. 252.)
S'(Ms) denotes the semigroup of operator endomorphisms of Ms. (§11.1,
p. 251; §11.7, p. 279.)
stf(Ms) denotes the group of operator automorphisms of Ms. (§11.1, P- 251;
§11.8, p. 281.)
FM denotes the set of fixed elements of Ms. (§11.5, p. 269.)
^-rad S denotes the ^-radical of S, which is the intersection of all congruences
a on S such that Sja is a semigroup of type ^ ;
means the derived type of $; S has type # ' if and only if ^-rad S = cs-
(§11.6, p. 275.)
rad S means */-rad S, where J is the type of right irreducible semigroups;
rad°$ is the (rad $)-class of S containing 0, and coincides with—
N(S), the nilradical of £. (§6.6, p. 38; §11.6, pp. 277, 278.)
c/T(p) means the normalizer in S of the right congruence p on 8; it is the sub-
semigroup {aeS: (s, t)ep implies (as, at)ep} ofS. (§11.6, p. 279.)
Let S be a cancellative semigroup satisfying the quotient condition Z
(§12.4, p. 297), and let a, beS.—
a/b denotes the right quotient {(#, y)eS x S:xa = yb) of a by b ;
a\b denotes the dually defined left quotient of a by b. (§12.4, p. 298.)
o(I) denotes the Malcev system of equations corresponding to the Malcev
sequence I. (§12.6, p. 310.)
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