NOTATION USED IN VOLUME ONE Square brackets are used for alternative readings and for reference to the bibliography. Let A and B be sets. A C B (or Bo A) means A is properly contained in B. A c B (or B 2 A) means A C B or A = B. A\B means the set of elements of A which are not in J5. Ax B means the set of all ordered pairs (a, b) with a in A, b in J3. The signs U and n are reserved for union and intersection, respectively, of sets and relations. The signs v and A will be used for join and meet in [semi]lattices. |-41 means the cardinal number of the set A. The sign o is used for composition of relations (§1.4), but is usually omitted for composition of mappings. denotes the empty set, mapping, or relation. i [iA] denotes the identity mapping or relation [on the set A], If ^ is a mapping whose domain includes A, then f\A means f restricted to A. {#!,• •, an} means the set whose members are ai,« •, an. Braces are sometimes omitted on single elements, for example Aub instead of A U {&}. If P(x) is a proposition for each element a: of a set X, then the set of all a in I for which P(x) is true is denoted by either {xeX: P(x)} or {x:P(x),xeX}. If M(x) is a set for each a in a set X, then the union of all the sets M(x) with x in X is denoted by either \JXexM(x) or \J{M(x) :XEX}. If F(x) is a member of a set C for each x in a set X, then the subset of C consisting of all F(x) with x in X is denoted by {F(x): x e X). If X = A x B, we may write {F(a, b):aGA,beB} instead of {F(a, b): (a, b)eAx B}. If A is a subset of a semigroup 8, then ^4 denotes the subsemigroup of 8 generated by A. If S is a group, then the subgroup of 8 generated by A is ( i u i - 1 ) , where A~l = { a ^ i a e i } . If ^4 and B are subsets of a semigroup 8, then ^4J5 means {ab:aeA, beB}. S1 [S°] means the semigroup /Sul[SuO ] arising from a semigroup S by the adjunction of an identity element 1 [a zero element 0], unless 8 already has an identity [has a zero, and \S\ 1], in which case S1 = 8 [S° = 8], (§1.1) a\b means "a divides b", that is, beaSl where a and b are elements of a commutative semigroup 8. (§4.3) pa [VI denotes the inner right [left] translation x-xa[x-ax] of a semigroup S, where a is a fixed element of 8. (§1.3) xiii
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