On injective and subdirectly irreducible S-acts over left zero semigroups

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

The aim of this paper is to characterize subdirectly irreducible S-acts over left zero semigroups. Also we compute the number of such acts and specify cogenerators acts over left zero semigroups. To do these we first take another look at the description of injective hulls of the separated S-acts over left zero semigroups.

Anahtar Kelimeler:

S-act, separated, injective, subdirectly irreducible, left zero semigroup

On injective and subdirectly irreducible S-acts over left zero semigroups

Keywords:

S-act, separated, injective, subdirectly irreducible, left zero semigroup,

PDF

___

We close the paper by characterizing simple S -acts. Recall that an S -act A is called simple if ConA = { , ∇}. It is easy to check that every S -act A with |A| ≤ 2 is simple but no S -act A with trivial action and |A| > 2 is simple.
Theorem 3.11 For a left zero semigroup S , there exists no simple S -act A with|A| > 2. Proof.
Let a = b be elements of A. Then in the case where a, b ∈ FixA we have ρa,b = ∇, since |A| > 2, hence there exists (a, b =)x ∈ A and (a, x) /∈ ρa,b. Therefore ρa,bis a non trivial congruence on A . Also in the case where one of a, b is not Şxed, taking a /∈ FixA, then ρa,b = ∇. Since otherwise if ρa,b=∇ then for each x = y ∈ A, we have (x, y) ∈ ρ
a,b. Therefore there exist s, t∈ S such that as = x, bt = y . Thus x, y ∈ FixA, by Lemma 1.1. Thus (a, x) /∈ ρx,y, and so ρx,yis a non trivial congruence on A . 2