tailieunhanh - On injective and subdirectly irreducible S-acts over left zero semigroups
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. | Turk J Math 36 (2012) , 359 – 365. ¨ ITAK ˙ c TUB doi: On injective and subdirectly irreducible S -acts over left zero semigroups Gholamreza Moghaddasi Abstract 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. Key Words: S -act, separated, injective, subdirectly irreducible, left zero semigroup 1. Preliminaries Recall that, for a semigroup S , a (right) S -act (or S - system) is a set A together with a function α : A × S → A, called the action of S (or the S -action) on A, such that for a ∈ A and s, t ∈ S (denoting α(a, s) by as), a(st) = (as)t, and if S is a monoid, with 1 as its identity, a1 = a. An S -act A is called separated if for a = b in A there exists an s ∈ S \ {1} with as = bs. A homomorphism f : A → B between S -acts is a function such that for each a ∈ X , s ∈ S , f(as) = f(a)s. We denote the category of all (right) S -acts and homomorphisms between them by Act-S. An element a of an S -act A is called a fixed or zero element if as = a for all s ∈ S . We denote the set of all fixed elements of an S -act A by FixA, which is in fact a subact of A. We will see that the set FixA plays an important role for acts over left zero semigroups. Notice that for a given S -act A, the set (FixA)S of all functions from S to FixA is an S -act. In fact, using the notation (as )s∈S for a given element of (FixA)S , (as )s∈S · t is defined to be the constant family (at )s∈S , for t ∈ S . Recall that a semigroup S all whose elements is a left zero of S is called a left zero semigroup. This class of semigroups is important and useful since: 1) Every non empty set S can be turned into a left zero semigroup by defining st = s for all s, t ∈ S . 2) The definition of rectangular .
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