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).