XIV
NOTATION
If p is an equivalence relation on a set X, and if (a, b) e p, then we write
a pb and say that a and b are p-equivalent, and that they belong to the
same p-class.
If p is a congruence relation on a semigroup 8, then 8/p denotes the factor
semigroup of S modulo /, and p* denotes the natural mapping of S upon 8/p.
(§1.5) 81 J denotes the Rees factor semigroup of S modulo an ideal J.
Let S be a semigroup, and let aeS. (Following from §2.1)
L(a) denotes the principal left ideal
Sxa,
R(a) denotes the principal right ideal
aSl.
J (a) denotes the principal two-sided ideal
S^-aS1.
& means {(a, b)eSxS: L(a) = £(&)}.
SI means {(a, b)e8xS: R(a) = R(b)}.
/ means {(a, b)eSxS :J{a) = J(b)}.
3tf means £c\0t.
2 means &m ( = 010.£?).
i
a
, P
a
, t/a, -Ha, Z?a mean respectively the &, ^ , ,/ , Jf, ^-class containing a.
/(a) means «/(a)\«/a. (It is empty or an ideal of S.)
J(a)/I(a) is the principal factor of S corresponding to a. (§2.6)
&x means the semigroup of all transformations of a set X. (§1.1)
&x means the group of all permutations of a set X. (§1.1)
*fx means the symmetric inverse semigroup on a set X. (§1.9)
@x means the semigroup of all binary relations on X. (§1.4)
!Fx means the free semigroup on X. (§1.12)
J^Sfr means the free group on X. (§1.12)
means the bicyclic semigroup. (§1.12)
JK^(0; /, A; P) means the Rees I xA matrix semigroup over the group with
zero G°, with Ax I sandwich matrix P.
JK{0; I, A; P) means the Rees I xA matrix semigroup without zero over
the group 0, with Ax I sandwich matrix P. (§3.1)
JS?^"(F) means the algebra of all linear transformations of a vector space V,
or the multiplicative semigroup thereof. (§§2.2, 5.1)
($l)n means the algebra of all n x n matrices over an algebra 51. (§5.1)
0[/S] means the algebra of a semigroup 8 over a field O. (§5.2)
£ means "isomorphic". (§1.3)
~ means "homomorphic", and sometimes "equivalent". (§1.3)
® is used for the direct sum of algebras, vector spaces, and representations.
(§5-1)
The nxn identity matrix is denoted by:
In when it is over a field (§§5.2, 5.3),
Un when it is over an algebra with identity element u (§5.1),
An when it is over a group with zero (§3.1).
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