有限可換準群の構成

その他

A semigroup G with kernel K is called a nilpotent semigroup if it satisfies the following condition (C. 1) :
(C. 1) G⊃G^2⊃G^3⊃. . . . . ⊃G^p = K for some positive integer p.
In particular, we shall call G a nilsemigroup if it is a nilpotent semigroup and its kernel is a subgroup of G. Further, a semigroup M with zero 0 is called a generalized nilsemigroup if M^* = M＼{0} constitutes a nilsubsemigroup of M. Let T be a semigroup, and let 0 be a symbol not representing any element of T. Extend the given binary operation in T to one in T∪{0} by defining 00=0 and a0=0a=0 for every a in T. It is easy to see that T∪{0} becomes a semigroup with zero element 0 with respect to this binary operation. We speak of the passage from the semigroup T to the semigroup T∪{0} as "the addition of a zero element to T ", and denote the semigroup T∪{0} by T^0. It is clear that if T is a nilsemigroup then T^0 is a generalized nilsemigroup, and conversely that every generalized nilsemigroup can be obtained by adding a zero element to a nilsemigroup. As defined by Tamura [5], a semigroup is called a z-semigroup if it has a zero element, 0, but has no idempotent except 0. In particular, for a finite semigroup S it can be easily proved by the Corollary to Lemma 2 of Tamura [4] that S is a z-semigroup if and only if it satisfies the following condition (C. 2) :
<tt>
┌ (1) S has a zero element 0,
(C. 2) ┤
└ (2) S⊃S^2⊃S^3⊃ . . . . . . ⊃S^p = {0} for some positive integer p.
</tt>
That is, S is a z-semigroup if and only if it is a nilsemigroup with zero.
Let A and B be commutative semigroups, B having a zero element 0. Then, sometimes an ideal extension C of A by B in the sense of Clifford [1] can be commutative. If C is commutative, then C is called a commutative ideal extension of A by B. It should be noted that there exists at least one commutative ideal extension of A by B. If G is a finite [commutative] nilsemigroup having a kernel K, then the Rees factor semigroup G/K of G mod K is a finite [commutative] z-semigroup. Therefore, we can say that G is a [commutative] ideal extension of a finite [commutative] group by a finite [commutative] z-semigroup.
In this paper, we shall present a method of constructing all possible finite commutative semigtoups by using the concepts introduced above.