Compressible S-system

Transcription

1 Compressible S-system Asaad M. A. Alhosaini Amna Salah Hasan Al-Kraimy Department of Mathematics, University of Babylon, Iraq. Abstract It is well known that an S-system over semigroup (monoid) is a generalization of module over a ring. So, many of the notions related to modules, were converted to S-system. In this paper the notion of compressible modules (and related concepts), will be convert to S-system, and investigate the corresponding properties and relations will be investigated. Keywords: Compressible S-system, critically compressible S-system, retractable S-system, weakly monomorphism, weakly compressible. الخالصة من المعموم ان انظمة S عمى شبو الزمرة )او المونويد( ىي تعميم لمموديوالت عمى الحمقة. لذلك كثير من المفاىيم المتعمقة بالموديوالت قد تم تحويرىا الى انظمة. S في ىذا البحث قمنا بتحوير مفيوم الموديوالت المضغوطة) والمفاىيم المرتبطة بيا( الى S انظمة S ودرسنا الصفات والعالقات المتعمقة بيذه المفاىيم. S الكممات المفتاحية: S واالنضغاط الضعيف انظمة المضغوطة, انظمة المضغوطة حرجا, انظمة القابمة لمسحب, التشاكل االحادي الضعيف, 1.Introduction and Preliminaries: Throughout this paper, let be a monoid. A unitary right S-system over which is denoted by is a non empty set with a function,( ) ( ) such that : and ( ) ( ), we call a right S-system or a right system over and write. More informally we often say that defines a right multiplication of elements from by elements of. Analogously, left S-system is defined and write. (see [ Kilp and Mikhalev, 2000 ] ). If has a zero element 0, then is a fixed element for all, and it is denoted by.(see [ Antonio, Lopez and John, 1979 ] ). Let.( is non empty subset of ) Here, is said to be a sub system of if for all and. ( see[ Kilp and Mikhalev, 2000 ] ). Let be an S- system, an element is called a zero of (a fixed element,a sink) if for all * + is a one element subsystem. If the monoid has a left zero, then every element for is a zero of. Every S-system can be extended to an S-system with a fixed element. by taking the disjoint union : * + ( see [ Kilp and Mikhalev, 2000 ] ). We call a simple system if it contains no sub system other than and one element sub system. ( see [ Kilp and Mikhalev, 2000 ] ). Let be two right S-systems. A function is called a homomorphism of S-system or an S-homomorphism if for each ( ) ( ) ( see [ Hinkle, 1974] ). The collection of all such S- homomorphism from will be denoted by ( ) or sometimes by ( ) ( see[ Hinkle, 1974] ).

2 If there exists ( ) which is monomerphism and epimorphism,we say is S-isomorphic to and write. ( see [ Hinkle, 1974] ). The identity mapping is a homomorphism of right S-system. The same is true for left S-system. (see [ Kilp and Mikhalev, 2000 ] ). Note that if is an S-homomorphism then ( ) ( ) is a sub system of. (see [ Kilp and Mikhalev, 2000 ] ). Let be an S-system. An equivalence relation on is called an S-system congruence or a congruence on,if implies ( ) ( ) for (see [ Kilp and Mikhalev, 2000 ] ). The identity S- congruence on will be denoted by. ( see [ Kilp and Mikhalev, 2000 ] ). Let M S be an S-system and be a congruence on M S. Define a right multiplication by element of S on the factor = {[m- m M} by [m- s = [ms- for every s S. From the property of an S-system congruence it follows at once that this multiplication is well-defined. As a result M S / becomes a right S-system which is called the factor S-system of M S by. The canonical surjection where (, - ) and is a congruence on an S-system M S, is a homomorphism which is called a canonical epimorphism.( see [ Kilp and Mikhalev, 2000 ] ). ( Homomorphism Theorem for S-system ) Let be an S-homomorphism and be a congruence on such that implies ( ) ( ) Then with (, - ) ( ) is the unique S-homomorphism such that the following diagram is commutative M s π ρ f M s ρ N s fˋ If, then is injective, and if is surjective, then is surjective. ( see [ Kilp and Mikhalev, 2000 ] ). Recall that is a simple S system if it contain no subsystem other than itself. Most of the modules notions, were reversed to S-system(S-acts).In this work we attempt to reverse this concept and related properties and relations to S-system.One of their notions is a compressible module that was investigated by many authors' ( see [Abhay, 2000 ], [ Virginia and Alveri, 2009 ] and [ Limarenko, 2006 ] ). A module is called compressible if it can be embedded in any of its nonzero submodules. ( see [ Virginia and Alveri, 2009 ] ). A compressible module is called critically compressible if it cannot be embedded in any proper factor module. ( see [ Virginia and Alveri, 2009 ] ). A partial endomorphism of is a homomorphism from a submodule of into. (see [ Abhay, 2000 ] ).

3 Proposition: Let be a compressible module. Then the following conditions are equivalent: a) is critically compressible ; b) Every nonzero partial endomorphism of is a monomorphism. ( see [ Virginia and Alveri, 2009 ] ). An -module is retractable (or slightly compressible) if ( ) for every nonzero submodule ( ) It is clear that compressible modules are retractable, but the converse is not true. ( see [ Virginia and Alveri, 2009 ] ). Proposition: Suppose that is a retractable -module. If every nonzero ( ) is a monomorphism, then every nonzero element of ( ) is a monomorphism, for any nonzero submodule of. In particular, is compressible. (see [ Virginia and Alveri, 2009 ] ). Proposition : Let be a retractable module. The following statements are equivalent : a. is critically compressible. b. Every nonzero partial endomorphism of is a monomorphism. ( see [Virginia and Alveri, 2009] ). A module is said to be fully retractable if for every nonzero submodule of and every nonzero element ( ) we have ( ). Clearly, if is fully retractable then it is retractable. (see [Virginia and Alveri, 2009]). Proposition: A compressible right module is simple if it satisfies one of the following conditions: 1. The module is finite ; 2. The module has a finite number of submodules ; 3. The module has a nonzero finite submodule ; 4. The module has a simple submodule. ( see [ Limarenko, 2006 ] ). A nonzero - submodule of a module is called rational (or dense) in if ( ), for any ( see [ Virginia and Alveri, 2009 ] ). An -module is called monoform if every submodule is dens. (see [ Virginia and Alveri, 2009 ] ). 2.Compressible S-system: (2.1) Definition :An S-homomorphism where is a subsystem of is called a partial endomorphism of. (2.2) Definition : An S-system is called compressible if it can be embedded in each of its nonzero sub systems. (, there is a monomorphism ). In the following, some of the properties of compressible S-system are given. (2.3) Proposition: A compressible S-system is simple if it satisfies one of the fallowing conditions : 1. is finite ; 2. has a finite number of sub systems ; 3. has a non zero finite sub system ; 4. has a simple sub system. Proof : The first, second and third condition are particular cases of the fourth condition. 4. Assume that is a compressible S-system, and let be a simple sub system of, then monomorphism

4 ( ). But ( ) (since is simple ). Therefore, that is is simple. 1. Let be finite has a minimal sub system (which is simple ) then by (4 ) is simple. 2. Assume that has a finite number of sub systems has a minimal sub system (which is simple ) then by (4 ) which also is simple. 3. Assume that has a non zero finite sub system, then monomorphism then is finite, By (1) is simple. (2.4) Proposition : Any subsystem of a compressible S-system is compressible, while a homomophic image of a compressible S-system need not be compressible. Proof : Assume is compressible S-system and then a monomorphism, hence : is a monomorphism, too. That is can be embedded in any of its non-zero subsystem. For the second part of the remark, see Example (2.7). (2.5) Definition: A compressible S-system is called critically compressible if it cannot be embedded in any of its proper factor system. (2.6) Proposition : Any subsystem of critically compressible S-system is critically compressible, too. Proof : Assume that is critically compressible and then, by Remark (2.4). is compressible, if is a congruence on and is a monomorphism, since is compressible there is, we have is a congruence on (it is an extension of to ), and can be considered as a subsystem of with as the inclusion map. In this case is a monomorphism, contradicts the assumption that is critically compressible. Therefore is critically compressible. (2.7) Example : Let ( ) and * + is an S-system, and define. Any non-zero sub system of is of the form * + where (In fact has no zero subsystem ). Let defined by ( ), then is a homomorphism, ( ) ( ) ( ) ( ) - let in then ( ) ( ) therefore is a onomorphism. That is is a compressible S-system., that is any nontrivial factor of is finite,while is an infinite system. So, cannot be embedded in its proper factors, hence is critically compressible S-system. On the other ( ) is a homomophic image of, which is finite and not simple, and so, not compressible. In the following are given an analogous result to the case of modules. (2.8) Proposition: The following conditions are equivalent for a compressible S- system : i. is critically compressible ; ii. Every nonzero partial endomorphism of is a monomorphism. Proof: i ii Assume that is critically compressible, and let be a partial endomorphism ( is a sub system of )

5 then, ( for simplicity we write ) since is compressible monomorphism. Let *( ) and ( ) +, Indeed, it is a congruence on [ is an extension of the congruence to ( i.e. ) ] define,, -, - (it is clear that is monomorphism ), then is a monomorphism By (i) is not proper, that is,,which implies. Therefore is monomorphism. Assume that is a homomorphism, where is a congruence on,. Assume that and let be the natural epimorphism. Let, such that ( ) ( ) Therefore is an isomorphism, then is a partial endomorphis. which is not a monomorphism (since is not proper then such that and ( ) ( ) then ( ( )) ( ( )),which is a contradiction. Therefore is not a monomorphism. To introduce the next concept (retractable S-system )which is weaker than compressible property,we first need the following remark. (2.9) Remark: Let be two S-systems and has at most one zero. Then ( ) if and only if. Proof : Let ( ), then it is clear that either or. Let ( ) with, then for some, ( ), take such that ( ) ( ) then let [ ( ) ( ) -, in this case ( ) Hence ( ). (2.10) Definition: An S-system is retractable(or slightly compressible) if ( ) for every non-zero subsystem of ( ) Khosravi in (2012), introduced,the notion of retractable S-system, using the condition ( ), for each subsystem of.we preferred the condition in Definition (2. 10), because it excludes the case ( ), too. By the Khosravi`s definition, any S-system over a monoid with zero is retractable. It is clear that every compressible S-system is retractable but the converse is not true. For example : Consider the monoid S ( ), A={0,2,3,4}, is an S-system by as a multiplication, the proper subsystems of are * + * + * + only ( ), ( ) It is clear that Hence is retractable, but not compressible.(since is finite and not simple ). The following concept is needed in some of the next results. (2.11) Definition : Let be an S system, with the property that and imply then we say ( ) is a zero divisor free (for short ZDF) monoid.

6 (2.12) Remark: It is clear that, if has no zero element then is a ZDF monoid. (2.13) Proposition: Suppose that is a retractable S-system if every nonzero ( ) is a monomorphism, then every nonzero element of ( ) is a monmorphism, for any nonzero subsystem of In particular, is compressible. Proof : Let ( is retractable) and ( ) be the inclusion map. Then is monomorphism So that a monomorphism. (2.14) Proposition : Let be a retractable S-system. The following statements are equivalent : i. is critically compressible ; ii. every nonzero partial endomorphism of is a monomorphism. Proof : It is clear from Proposition (2.8) Since every endomorphism of is a partial endomorphism, and hence is monomorphisim, then is compressible too by Proposition (2.13), therefore by Proposition (2.8) is critically compressible. Analogous to the notions of rational submodule, monoform and uniform modules, in the following we introduce similar (and some different ) notions in S- system. (2.15) Definition: A subsystem of an S- system is called rational in if ( ) (or empty) for every sub system of with. (2.16) Definition: An S-system is called monoform if every subsystem is rational. In module theory, a module is monoform if and only if every partial endomorphism is a monomorphism. This is not the case in S-systems, in the following we discuss, the relation between the two properties in details. (2.17) Lemma: If any partial endomorphism of an S-system is injective, then is monoform. Proof : Assume that is not monoform, then it has a subsystem, say, which is not rational, that is, is contained in a subsystem, and ( ). Let be a homomorphisom with (it is clear that ) Let be the natural epimorphism then is a partial endomorphism, which is not injective. This contradicts the assumption. Therefore any subsystem of is rational, and is monoform. The converse of the above lemma is not true, as in the following example. (2.18) Example: Let ( ) with usual multiplication (in, and for the elements of by the elements of ). is monoform, but defined by ( ) is an endomorphism (and so a partial endomorphism) which is not injective. (2.19) Definition: Let be an S-homomorphism, is said to be weakly monomorphism if for each non-zero subsystem of, ( ). (2.20) Remark: Any monomorphism is weakly monomorphism, since if * +, then it has more than one element. Hence ( ) means ( ) ( ) for some in The converse is not true. In Example (2.18), is weakly monomorphism, but not a monomorphism.

7 (2.21) Lemma: is a monoform S-system, if and only if every nonzero partial endomorphism of is weakly monomorphism. Proof : Assume that is a subsystem of and is not a weakly monomorphism. Then contains (properly) a subsystem, such that ( ).Then is defined by (, -) f(x) x N x N is an S-homomorphism since ( ) (, - ) (, -) ( ) ( ) (, -) ( ) ( ) case 1: (, - ) (, -) while (, -) ( ). case 2: x (, -) and (, -) while (, - ) (, -). In the two cases (, - ) (, -). Therefor is a homomorphism. Since and there is such that ( ), then (, -) ( ). That is. Therefore is not rational in, and so, is not monoform. Conversely assume that is not a monoform, and is not rational in. Let be such that, and is a non zero S- homomorphism, then consider the composition Where is the natural epimorphism of onto then is a non-zero partial endomorphism of, which is not a weakly monomorphism, since ( ) (2.22) Proposition: Let be an S-system, consider the following i. is a monoform ; ii. every partial endomorphism of is monomorphism ; iii. every partial endomorphism of is a weakly monomorphism. Then : (i) if and only if (iii). And (ii) implies (iii). Proof : (i) (iii) By Lemma (2.21) and Remark (2.20). (ii) (iii) By Remark (2.20). Note that (iii) does not imply (ii) (see Example (2.18)). (2.23) Remark: is a weakly monomorphism, then is a weakly monomorphism. Assume ( ) then ( ( )) (2.24) Lemma: If is retractable, then ( ) is ZDF, if and only if for each ( ) is a weakly monomorphism. Proof : Let ( ) and ( ) for some since is retractable, there is so (where is the inclusion map of into ) is a non zero endomorphism of, denote Note that ( ), hence contradicts the hypothesis that ( ) is ZDF.

8 Let ( ), both nonzero, then both are weakly monomorphism, hence ( ( )) that is. Therefore ( ) is ZDF. (2.25) Definition: An S-system is weakly compressible if for each there is a weakly monomorphism. It is clear that any compressible S-system is weakly compressible. (2.26) Proposition: Let be a retractable S-system. If ( ) is ZDF, then is weakly compressible. Proof: Let, since is retractable, there is, and ( ), then by Lemma (2.24), is a weakly monomorphism, then by Remark (2.23), is a weakly monomorphism. Hence is weakly compressible. Reference Abhay. K. S."Compressible Module", Mathematics subject Classification", (2000), pp:1-11. Antonio. M and John.K."Quasi-injective S -systems and their S-endomorphism Semigroup", Czechoslovak Mathematical Journal, (1979), pp: , Vol: 29, No: 1. Hinkle. C. V."The Extended Centralizer of an S-Set", pacific Journal of Mathematics, (1974), pp: , Vol:53, No:1. Khosravi. R. "On Retractable S-Acts", Journal of Mathematical Research with Applications, (2012), pp: , Vol:32, No:4. Kilp. M, Knauer. U and Mikhalev. A. "Monoids, Acts and Categories", Walter de Gruyter, Berlin, New York. ( 2000), pp:1-70. Limarenko. S. V. "Weakly Primitive Superrings", jornal of Mathmetical science, (2006), pp: , Vol :139, No:4. Virginia.S. R. and Alveri.A. S."A Note on a Problem due to Zelmanowitz", Algebra and discrete Mathematics, (2009), pp:85-93, No:3.