On idempotent generated semigroups

0.06 MB·English

by  João Araújo

On Idempotent Generated Semigroups
1
1
0
2
Jo˜ao Arau´jo

n
a
J

9 Abstract
2

We provide short and direct proofs for some classical theorems
]
R proved by Howie, Levi and McFadden concerning idempotent gener-
G ated semigroups of transformations on a finite set.
. Mathematics subject classification: 20M20.
h
t
a
m Let n be a natural number and [n] = {1,...,n}. Let Tn,Sn be, respec-
tively, the transformation semigroup and the symmetric group on [n]. Let
[

1 P = (Ai)i∈[k] be a partition of [n] and let C = {x1,...,xk} be a cross-
v section of P (say xi ∈ Ai). Then we represent Ai by [xi]P and the pair
9 (P,C) induces an idempotent mapping defined by [x ] e = {x }. Con-
0 i P i
7 versely, every idempotent can be so constructed. To save space instead of
5 [x ] ... [x ]
1. e = x1 P ... xk P we write e = ([x1]P,...,[xk]P). This notation ex-
1 k !
0
tends to e = ([x ,y] ,[x ] ,...,[x ] ) when y ∈ [x ] and [x ] e = {x }. By
1 1 P 2 P k P 1 P i P i
1 ([x ],...,[x ,y],...,[x ]) we denote the set of all idempotents e with image
1 i k
:
v {x ,...,x } and such that the Ker(e)-class of x contains (at least) two ele-
1 n i
Xi ments: xi and y, where the underlined element (in this case xi) is the image
r of the class under e. For a ∈ Tn, we denote the image of a by ∇a.
a
Lemma 1 Let a ∈ I = T \ S , rank(a) = k, and (xy) ∈ Sym(X).
n n n
Then a(xy) = ab, where b = 1 or b = e e e , with e2 = e and rank(e ) = k.
1 2 3 i i i
Proof. Let ∇a = {a | i ∈ [k]}. If x,y 6∈ ∇a, then a(xy) = a. If, say,
i
x = a and y 6∈ ∇a, then ae = a(a y), for all e ∈ ([a ,y],[a ],...,[a ]).
1 1 1 1 1 2 k
Finally, if x,y ∈ ∇a, without loss of generality, we can assume that x = a
1
and y = a . Then a(a a ) = ae e e , for e ∈ ([a ],[a ,u],[a ],...,[a ]),
2 1 2 2 3 4 2 1 2 3 k
e ∈ ([u],[a ,a ],[a ],...,[a ]) and e ∈ ([u,a ],[a ],...,[a ]).
3 1 2 3 k 4 1 2 k
1
Theorem 2 [1] Every ideal of I is generated by its own idempotents.
n
Proof. Let a ∈ I . Then a = eg for some e = e2 ∈ T and g ∈ S . Therefore
n n n
a = e(x y )...(x y ) and hence, applying m times Lemma 1, a can be
1 1 m m
obtained as a product of idempotents of the same rank as a.
Let t ∈ I and g ∈ S . Denote g−1tg by tg and let C = {tg | g ∈ S } and
n n t n
tSn = h{t}∪S i\S . Forf = ([x ,w] ,[x ] ,...,[x ] ) andg ∈ S , it is easy
n n 1 Q 2 Q k Q n
to check that we have fg = ([x g,wg] ,...,[x g] ) ∈ ([x g,wg],...,[x g]).
1 Qg k Qg 1 k
Lemma 3 Let a ∈ I , rank(a) = k, let f = ([x ,w] ,[x ] ,...,[x ] )
n 1 Q 2 Q k Q
and let (xy) ∈ S . Then a(xy) = ab, where b = 1 or b = e e e , with e ∈ C .
n 1 2 3 i f
Proof. AsinLemma1,oneonlyhastoshowthatthesets([a ,y],[a ],...,[a ]),
1 2 k
([a ],[a ,u],...,[a ]), ([u],[a ,a ],...,[a ]) and ([a ,u],[a ],...,[a ]) inter-
1 2 k 1 2 k 1 2 k
sect C . Let g ∈ S such that x g = a ,wg = y and x g = a (2 ≤ i ≤ k).
f n 1 1 i i
Thus fg = ([x g,wg] ,[x g] ,...,[x g] ) ∈ ([a ,y],[a ],...,[a ]). The
1 Qg 2 Qg k Qg 1 2 k
proof that C intersects the remaining three sets is similar.
f
Repeating the arguments of Theorem 1 together with Lemma 3 we have
Corollary 4 Let eg ∈ I (for some e = e2, g ∈ S ) and let f2 = f with
n n
rank(eg) = rank(f). Then eg ∈ ehC i and, in particular, eg ∈ hC i.
f e
From now on let a = eg ∈ I (for some e = e2, g ∈ S ).
n n
Corollary 5 hC i = eSn = (eg)Sn = aSn.
e
Proof. The only non-trivial inclusion is eSn ⊆ hC i. As eSn is generated by
e
{geh | g,h ∈ Sn}, let g,h ∈ Sn. By Corollary 4, he = (heh−1)h ∈ hCheh−1i
and eg ∈ hCei. Since heh−1 ∈ hCei, it follows that hCheh−1i ≤ hCei. Thus
heg = (he)(eg) ∈ hC i. It is proved that eSn ⊆ hC i.
e e
Theorem 6 [2] hC i = hC i = aSn and hence is idempotent generated.
a e
Proof. Since a = eg, then e = ag−1 and it is easy to check that we have
(ag−1)n = aagag2 ...agn−1g−n. But for some natural m we have g−m = 1.
Thus (ag−1)m = aagag2 ...agm−1 ∈ hC i so that e = em = (ag−1)m ∈ hC i
a a
and hence, by Corollary 5, it follows that aSn = hC i ≤ hC i ≤ aSn. (Cf.
e a
[3]).
2
References
[1] J. M. Howie, R. B. McFadden Idempotent rank in finite full transformation semi-
groups, Proc. R. Soc. Edinburgh, Sect. A 114 (1990), 161–167.
[2] I. Levi and R. McFadden Sn-normal semigroups, Proc. Edinburgh Math. Soc., 37
(1994), 471-476.
[3] D.B.McAlisterSemigroups generatedbyagroupandanidempotent,Comm.Algebra,
26 (1998), 515-547.
3