Subpullbacks and Po-flatness Properties of S-posets

Transcription

1 Journal of Sciences, Islamic Republic of Iran 25(4): (2014) University of Tehran, ISSN Subpullbacks and Po-flatness Properties of S-posets A. Golchin * and L. Nouri Department of Mathematics, Faculty of Sciences,University of Sistan and Baluchestan, Zahedan, Islamic Republic of Iran Received: 16 April 2014 / Revised: 1 November 2014 / Accepted: 19 November 2014 Abstract In (Golchin A. and Rezaei P., Subpullbacks and flatness properties of S-posets. Comm. Algebra. 37: (2009)) study was initiated of flatness properties of right -posets over a pomonoid that can be described by surjectivity of corresponding to certain (sub)pullback diagrams and new properties such as () and () were discovered. In this article first of all we describe poflatness properties of -posets over pomonoids by po-surjectivity of corresponding to certain subpullback diagrams. Then we introduce three new Conditions ( ), () and () and investigate the relation between them and Conditions (), ( ), (), (), () and (). Also we describe these properties by po-surjectivity of corresponding to certain subpullback diagrams and describe Conditions ( ), () and () by weak po-surjectivity of corresponding to certain subpullback diagrams. Keywords: Ordered monoid; -poset; Subpullback diagram. Introduction In [5] the notation (,,,, ) was introduced to denote the pullback diagram of homomorphisms : S M S Q and : S N S Q in the category of left -acts, where is a monoid. Tensoring such a diagram by produces a diagram (in the category of sets) that may or may not be a pullback diagram, depending on whether or not the mapping, obtained via the universal property of pullbacks in the category of sets, is bijective. It was shown that, if we require either bijectivity or surjectivity of for pullback diagrams of certain types, we not only recover most of the wellknown forms of flatness, but also obtain Conditions () and () as well. In [4] Golchin and Rezaei extended the results from [5] to -posets and two new Conditions () and () were introduced. Let be a pomonoid. A poset is called a right poset if acts on in such a way that () the action is monotonic in each of the variables, () for all, and, () = () and 1 =. Left -posets are defined similarly. The notations and S B will often be used to denote a right or left -poset and Θ ={} is the one-element right -poset. The -poset morphisms are the order-preserving maps that also preserve the action. We denote the category of all right -posets, with -poset maps between them, by Pos-. We recall from [3] that an -poset map : is called an embedding if () ( ) implies, for,. We recall from [2] that in the categories of -posets and posets the order relation on morphism sets is defined pointwise (i.e. for, : if and only if () () for every ). In such * Corresponding author: Tel: ; Fax: , agdm@math.usb.ac.ir 369

2 Vol. 25 No. 4 Autumn 2014 A. Golchin and L. Nouri. J. Sci. I. R. Iran categories, a diagram is subcommutative if. Iff and for every left -poset and all homomorphisms : and : such that, there exists a unique homomorphism h: such that h= and h=, then diagram ( ) willl be called subpullback diagram for and. In the category of -posets or the category of posets, may in factt be realized as = {(, ) () ()} where and are the first and second coordinate projections. (Notice that is possibly empty.) The subpullback diagram ( ) will be denoted by (,,,,, ). Tensoring the subpullback diagram (,,,,, ) by any right -poset one gets the subcommutative diagram in the category of posets. For thee subpullbackk of mappings and in the category of posets we may take = {(, ) ( ) ( ) () ()} with and the restrictions of the projections. It follows from the definition of subpullbacks that there exists a unique monotone mapping : such that, in the diagram we have = for =1,2. We shall calll this mapping the corresponding to thee subpullback diagram (,,,,). It can be seen that the mapping in diagram ( ) is given by ( (, )) = (, ) for all and (, ) S P. we define po-surjectivity and weak po-surjectivity of corresponding to thee subpullback diagram (,,,, ) as (, )( ) S M)( S N) () () ( )( S M)( S N)(( ) ( ), =, )] and (, )( ) S M)( S N) () () ( )( S M)( S N) (( ) ( ),, )], respectively. Obviously, surjectivity s off po-surjectivity of weak po-surjectivity of f. We say that an -poset satisfies Condition ( ) if for all, and,, implies =,, for some and,,, such that. An -poset satisfies Conditionn () if for all elements,, all homomorphismss : S( ) S S and all,, if () (), then there exist,,,, {,}, such that ( ) ( ) and =, in S ( ). An -poset satisfies Condition () if for all, and, implies =,, for some and a,, such that. For a pomonoid the following relations exist among flatness properties p of an -poset. 370

3 Subpullbacks and Po-flatness Properties of S-posets free projective () () () ( ) () () ( ) () () w. po f () w. f ( ) In this article for a pomonoid, using the information from [5, 4] we describe po-flatness and Conditions ( ), () and () of -posets according to po-surjectivity of corresponding to certain subpullback diagrams. We also describe Conditions ( ), () and () introduced in [7, 4] by weak po-surjectivity of corresponding to certain subpullback diagrams. Results 1. Po-flatness of -posets In this section for a pomonoid we give equivalences of po-flatness of posets by posurjectivity of the mapping corresponding to certain subpullback diagrams. Recall from [7] that an -poset is called po-flat (in Pos- ) if and only if, for all embeddings S B S C in the category of left -posets, the induced orderpreserving map SB SC is embedding, that is, in SC implies in SB. The -poset is called (principally) weakly po-flat if the functor preserves embeddings of (principal) left ideals of monoid into, that is, for all (principal) left ideals of monoid,,,,, in S S implies in S I. For more po-flatness properties we refer the reader to [8]. Lemma 1.1. [7, Lemma 3.5] An -poset is po-flat if and only if for every left -poset S B, any, and any, S B, in SB implies in S (Sb ). Theorem 1.2. An -poset is po-flat if and only if the corresponding is po-surjective for every subpullback diagram (,,,,), where : S M S Q is an embedding of left -posets. Proof. Suppose is po-flat, : S M S Q is an embedding of left -posets and let () ( ) in SQ, for, and, S M. Since is po-flat, is embedding, and so in S M. Since () (), = and ', the corresponding is posurjective for the subpullback diagram (,,,, ). Conversely, suppose the corresponding is posurjective for every subpullback diagram (,,,, ), where : S M S Q is an embedding of left -posets. Let S M be an -poset,,,, SM and in S M. If : S (Sm ) S M is the inclusion map, then is an embedding (since S(Sm ) is a -subposet of SM). Also () = = ( ). By posurjectivity of for the subpullback diagram (,,,, ), there exist " and, S(Sm ) such that () ( ), = " and " in S (Sm ). Thus, and so =" " in S (Sm ). That is, is po-flat as required. Similarly, we can prove the following theorems. Theorem 1.3. An -poset is weakly po-flat if and only if the corresponding is po-surjective for every subpullback diagram (,,,, ), where is a left ideal of and : S I S S is an embedding of left -posets. Theorem 1.4. An -poset is principally weakly po-flat if and only if the corresponding is posurjective for every subpullback diagram (,,,, ), where and : S (Ss) S S is an embedding of left -posets. We recall from [1] that an element of a pomonoid is called right po-cancellable if implies, for all,. An -poset is called potorsion free if implies, whenever, and is a right po-cancellable element of. Theorem 1.5. An -poset is po-torsion free if and only if the corresponding is po-surjective for every subpullback diagram (,,,, ), where : S S S S is an embedding of left -posets. Proof. Suppose is po-torsion free and let : S S SS be an embedding of left -posets. Then (1) is a right po-cancellable element of, for if (1) (1), for,, then () ( ), and so. Suppose () ( ) in S S, for, and,. Then () ( ), and so (1) (1). Since (1) is a right po-cancellable element of and is po-torsion free, we have, and so in SS. Since () (), = and, is po-surjective for the subpullback diagram (,,,, ). Conversely, suppose the corresponding is posurjective for every subpullback diagram (,,,, ) where : S S S S is an embedding of left -posets. Let be a right po-cancellable element of and let 371

4 Vol. 25 No. 4 Autumn 2014 A. Golchin and L. Nouri. J. Sci. I. R. Iran, for,. If : S S S S is defined by () =, for every, then is an embedding, also (1) = = (1) in S S. By po-surjectivity of for the subpullback diagram (,,,, ), there exist " and,, such that () (), 1=" and " 1 in S S. Thus, and so. Also, = " and ". Hence = " ", and so is po-torsion free. 2. -Posets satisfying Condition (( )) ( ) In this section we give some equivalent conditions on an -poset satisfying Condition (( )) ( ) according to (weak) po-surjectivity of corresponding to certain subpullback diagrams. Note that for -posets and S B, in S B, for, and, SB if and only if there exist,,,,,, S B,,,,,, such that. Lemma 2.1. An -poset satisfies Condition ( ) if and only if for all S B and all,,, SB, in S B implies the existence of " and,, such that = ", " and. Proof. Suppose satisfies Condition ( ) and let in S B, for, and, SB. Then there exist,,,,,, S B,,,,,, such that. Since and satisfies Condition ( ), there exist and,, such that =, and. Then ( )= ( ) ( ) = ( ) ( ) = ( ) ( ), and so we have the following -tossing of length 1 ( ) ( ) ( ) ( ). Continuing this procedure we have:. Again implies the existence of and,, such that =, and. Since and, we have ( ), and so there exist " and,, such that =", and ( ). If = then = ="( )= ", " and = ( ) ( )= ( ) ( ). The converse is obvious. Theorem 2.2. For an -poset the following (1) the corresponding is po-surjective for every subpullback diagram (,,,, ); (2) the corresponding is po-surjective for every subpullback diagram (,,,, ); (3) the corresponding is po-surjective for every subpullback diagram (,,,, ), where is a left ideal of ; (4) the corresponding is po-surjective for every subpullback diagram (,,,, ), ; (5) the corresponding is po-surjective for every subpullback diagram (,,,, ); (6) the corresponding is po-surjective for every subpullback diagram (,,,, ); (7) satisfies Condition ( ). Proof. Implications (1) (2) (3) (4) (5) and (2) (6) are obvious. (5) (7). Suppose, for, and,. Then. If () = and () =, for, then (1)= = (1). By assumption there exist " and,, such that () (), 1= " and " 1. Thus, =" and ", and so satisfies Condition ( ) as required. (6) (7). Suppose, for,,, and let S F = {, }, where (, ) = (, ) and (, ) (, ) if and only if s and =, for, and, {, }. Define : S F, as (1, ) = and (1, ) =. Since, we have, and so (1, ) = = (1, ). By assumption there exist " and (, ), (, ) SF, such that (, ) (, ), (1, ) = (, ) and " (, ) (1, ). 372

5 Subpullbacks and Po-flatness Properties of S-posets Then (1, ) = " (, ) implies that there exist,,,,,,,,,,,,,,, and (, ),...,(, ), (, ),, (, ) S F, such that (1, ) (, ) (, ) (, ) " (, ) (, ), " (, ) (, ) (, ) (, ) (, ) (1, ). Thus = =...= =, and so =. Also = =...= =, and so 1=", that is, = ". Similarly, " (, ) (1, ) implies that = and ". Since (, ) (, ), we have, and so satisfies Condition ( ) as required. (7) (1). Suppose () (), for,, S M and S N. By Lemma 2.1 there exist " and, such that = ", " and () (). Thus () (). If = and =, then ( ) ( ), = = = and = =. Thus the corresponding is po-surjective for every subpullback diagram (,,,, ) as required. Recall from [7] that an -poset satisfies Condition ( ), if, for, and, implies that there exist " and,, such that, and. Using an argument similar to that of the proof of Lemma 2.1 we have the following lemma. Lemma 2.3. An -poset satisfies Condition ( ) if and only if for all S B and all,,, S B, in S B implies the existence of and,, such that, and. Using Lemma 2.3 and an argument similar to that of the proof of Theorem 2.2 we have the following theorem. Theorem 2.4. For an -poset the following (1) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ); (2) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ); (3) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ), where is a left ideal of ; (4) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ), ; (5) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ); (6) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ); (7) satisfies Condition ( ). 3. -posets satisfying Condition (() ) () In this section first of all we give some equivalent conditions on an -poset satisfying Condition (() ) () according to (weak) po-surjectivity of corresponding to certain subpullback diagram. Then we give a necessary and a sufficient condition for an -poset to satisfy Condition (). Theorem 3.1. An -poset satisfies Condition () if and only if the corresponding is posurjective for every subpullback diagram (,,,, ), where is a left ideal of. Proof. Suppose satisfies Condition (), SI S S be an -poset morphism and let () () in S S, for, and,. Then, () (). If = and h = then h() h(), and so by assumption there exist ",,, and, {,}, such that h( ) h( ), = and in S J. Clearly,,, ( ) = h( ) h( ) = ( ) and implies that = and in SI. Thus the corresponding is po-surjective for every subpullback diagram (,,,, ). Conversely, suppose the corresponding is posurjective for every subpullback diagram (,,,, ), : S (Ss ) S S is an -poset morphism, for, and let () (), for,. Then () () in S S. By po-surjectivity of for the subpullback diagram (,,,,), there exist ",, and, {, }, such that ( ) ( ), =" and " in S (Ss ). Thus satisfies Condition () as required. An -poset satisfies Condition () if for all,, all homomorphisms : S (Ss ) S S and all,, if () (), then there exist ",,,, {, } such that ( ) ( ) and ", " in S (Ss ) (see [4]). Using an argument similar to that of the proof of Theorem 3.l we can show the following theorem. Theorem 3.2. An -poset satisfies Condition () if and only if the corresponding is weakly po- 373

6 Vol. 25 No. 4 Autumn 2014 A. Golchin and L. Nouri. J. Sci. I. R. Iran surjective for every subpullback diagram (,,,,), where is a left ideal of. Theorem 3.3. An -poset satisfies Condition () if and only if for all,, all homomorphisms : S (Ss ) S S and all,, if () (), then there exist ",,,,,,,,,,,,,,,,,, such that either () () and ", = =" " = = or () () and = ", " =,, or () () and = ", " = " =,., Proof. Necessity. Let S (Ss ) be a homomorphism, for, and () (), for,. By assumption there exist ",, and, {, }, such that ( ) ( ), = " and. Then =" and a", and so there exist,,,,,,,,,,,,,,,,,,,,,,, and,,,,,,,, {, }, such that,,. If =, =, =, =, = and =, then there are three cases as follows: Case 1. =. In this case there exist {1,...,} and {1,, }, such that =, = = = = =, =, = = = = =. Then () () = ( ) ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) ( )= () (). Since satisfies Condition (), there exist and,, such that =, and () (). Since also = there exist and,, such that =, and. Thus ( ) ( ) and, = =. Also, = = and since satisfies Condition (), there exist and,, such that =, and. On the other hand, = and so there exist and,, such that =, and. 374

7 Subpullbacks and Po-flatness Properties of S-posets Thus = = as required., Case 2. =. In this case there exists {1,..., }, such that = and = = = =. Thus () () = ( ) ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) = ( ) ( ) () Since satisfies Condition (), there exist and,, such that =, and () (). Since also =, there exist and,, such that =, and. Thus () ( ) and =, =, as required. Case 3. = and =. Then () (), =" and ". Since = " and satisfies Condition ( ), there exist, and,,,, such that = ", " =. Since ", there exist and,, such that " =, Sufficiency. Suppose : S(Ss ) is a homomorphism, for, and let () (), for,. By assumption there are three cases as follows: Case 1. () () and ", = = ", " = =. Thus = = = = = = " =" and = = = = = = = = =, and hence = ". Since ", we have ", and so ". Thus satisfies Condition () as required. Case 2. () () and =", " =. Thus =" =" and = = = = = =, as required. Case 3. () () and = ", " = " =,. 375

8 Vol. 25 No. 4 Autumn 2014 A. Golchin and L. Nouri. J. Sci. I. R. Iran Thus = = = = and = = = =, and so =. Also, = = = =, as required. 4. -Posets satisfying Condition (() ) () In this section we give some equivalent conditions on an -poset satisfying Condition (() ) () according to (weak) po-surjectivity of corresponding to certain subpullback diagrams. Theorem 4.1. For an -poset the following (1) the corresponding is po-surjective for every subpullback diagram (,,,, ), where ; (2) the corresponding is po-surjective for every subpullback diagram (,,,, ); (3) satisfies Condition (). Proof. Implication (1) (2) is clear. (2) (3). Suppose, for, and. Then, in S S. If : S S S S is defined as ()=, for, then ( 1) = = (1). By po-surjectivity of for the subpullback diagram (,,,,), there exist and, S S, such that () ( ), 1=" and " 1 in SS. Thus, = " and ", and so satisfies Condition () as required. (3) (1). Suppose S S SS is an -poset morphism and let () ( ) in S S, for, and,,. Then () ( ), and so () (). Since satisfies Condition () there exist " and,, such that = ", " and () (). Thus () (). If = and =, then = = = " = ", and " = " = " =, and so the corresponding is po-surjective for the subpullback diagram (,,,,). An -poset satisfies Condition (), if, for, and, implies that there exist " and,, such that ", " and (see [4]). Using an argument similar to that of the proof of Theorem 4.1 we can prove the following theorem. Theorem 4.2. For an -poset the following (1) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ), where ; (2) the corresponding is weakly po-surjective for every subpullback diagram (,,,, ); (3) satisfies Condition (). 5. Relations of the properties In this section we demonstrate that with one possible exception (( ) po-flat), all of the implications in Figure ( ) are strict. The proof of the following theorem is clear. Theorem 5.1. Let be an ordered group. Then all -posets satisfy Condition ( ). Theorem 5.2. [4, Theorem 4.6.] For any pomonoid, there exists an -poset that does not satisfy Condition (). Recall from [7], [8], [1], [4], that an -poset satisfies Condition (), if, for, and,, implies that there exist " and,, such that =", " = and. An -poset is called flat (in SPOS) if and only if, for all embeddings S B S C in the category of left -posets, the induced order-preserving map S B S C is injective. An -poset is called weakly flat if the induced morphism is injective for all embeddings of left ideals into S S. A pomonoid is called weakly right reversible in case (], for all, (if is a subset of a poset, (]: = { for some } is the down-set of, or the order ideal generated by ). An -poset satisfies Condition () if the corresponding is surjective for every subpullback diagram (,,,, ), where is a left ideal of. Using [4, Theorem 6.2.] the following theorem is clear. Theorem 5.3. For a pomonoid the following (1) Θ satisfies Condition (); (2) Θ satisfies Condition ( ); (3) Θ satisfies Condition ( ); (4) Θ is po-flat; (5) Θ is flat; (6) Θ satisfies Condition (); (7) Θ satisfies Condition () ; (8) Θ satisfies Condition () ; 376

9 Subpullbacks and Po-flatness Properties of S-posets (9) Θ is weakly po-flat; (10) Θ is weakly flat; (11) is weakly right reversible. The crucial things we need for distinctness of the properties are () flat (from Example 3 of [5]), (). (from Theorem 5.3), ( ) ( ) (from Theorems 5.1 and 5.2) and ( ) () (from the following example). Example 5.4. (( ) () ) Let = {1,, } be the monoid with the following table and trivial order, and let = {,,, } be the set with the following order: := {(,, ), (, ), (, ), (, ), (, ),(,)}. We give the covering relation and the figure of as follows: = {(,),(,)}. = 1 Define * =, for all and.. 1 Indeed is ann -poset. We claim that does not satisfy Condition (). Otherwise, implies that there exist and,, such thatt =, and. Then = implies that = and = 1, and so for every,, which is a contradiction. It can be shown that satisfies Condition ( ) (see( [6] pages 121 and 122). References 1. Bulman-Fleming S., Gilmour A., Gutermuth D. and Kilp M., Flatness properties of S-posets.. Comm. Algebra. 34: (2006). 2. Bulman-Fleming S. and Laan V., Lazard's theorem t for S- posets. Math. Nachr. 278: (2005). 3. Bulman-Fleming S. and Mahmoudi M., The category of S- posets. Semigroupp Forum. 71: (2005). 4. Golchin A. andd Rezaei P., Subpullbacks and flatnesss properties of S-posets. Comm. Algebra. 37: (2009). 5. Laan V., Pullbacks and flatnesss properties of acts I. Comm. Algebra. 29: (2001). 6. Nouri L., On flatness properties of product acts and posets.. Ph.D Thesis, Zahedan (2014). 7. Shi X., Strongly flat f and po-flat S-posets. Comm. Algebra. 33: : (2005). 8. Shi X., Liu Z., Z Wang F. and Bulman-Fleming S., Indecomposable, projective, and flat S-posets. Comm. Algebra. 33: (2005). 377