 Advances in Pure Mathematics, 2012, 2, 149-168 http://dx.doi.org/10.4236/apm.2012.23022 Published Online May 2012 (http://www.SciRP.org/journal/apm) Copyright © 2012 SciRes. APM Lattice of Finite Group Actions on Prism Manifolds John E. Kalliongis1, Ryo Ohashi2 1Department of Mathematics and Computer Science, Saint Louis University, St. Louis, USA 2Department of Mathematics, King’s College, Wilkes-Barre, USA Email: kalliongisje@slu.edu, ryoohashi@kings.edu Received October 12, 2011; revised December 11, 2011; accepted December 22, 2011 ABSTRACT The set of finite group actions (up to equivalence) which operate on a prism manifold M, preserve a Heegaard Klein bottle and have a fixed orbifold quotient type, form a partially ordered set. We describe the partial ordering of these actions by relating them to certain sets of ordered pairs of integers. There are seven possible orbifold quotient types, and for any fixed quotient type we show that the partially ordered set is isomorphic to a union of distributive lattices of a certain type. We give necessary and sufficent conditions, for these partially ordered sets to be isomorphic and to be a union of Boolean algebras. Keywords: Finite Group Action; Prism 3-Manifold; Equivalence of Actions; Orbifold; Partially Ordered Set; Distributive Lattice 1. Introduction This paper examines the partially ordered sets consisting of equivalence classes of finite group actions acting on prism manifolds and having a fixed orbifold quotient type. For a fixed quotient type, we show that the partially ordered set is a union of distributive lattices of a certain type. These lattices have the structure of factorization lattices. The results in this paper relate to those in , where those authors study a family of orientation revers- ing actions on lens spaces which is partially ordered in terms of a subset of the lattice of Gaussian integers or- dered by divisibility (see also ). Finite group actions on prism manifolds were also studied in . Let M be a prism manifold and let G be a finite group. A G-action on M is a monomorphism :DiffGM where DiffM is the group of self-diffeomorphisms of M. Two group actions :DiffGM and :G DiffM are equivalent if there is a homeo-morphism :hM M such that  1GhGh, and we let  denote the equivalence class. If :DiffGM is an action, let :MM be the orbifold covering map. The set of equivalence classes of actions on prism manifolds forms a partially ordered set by defining  if there is a covering :MM such that . A prism manifold is defined as follows: Let 11TSS be a torus where 1:1Sz z  is viewed as the set of complex numbers of norm 1 and 0, 1I. The twisted I-bundle over a Klein bottle is the quotient space ,, ,,1WTIuvtuv t. Let D2 be a unit disk with 21DS and let 12VSD be a solid torus. Then the boundary of both V and W is a torus 11SS. For relatively prime integers b and d, there exist integers a and b such that 1ad bc . The prism manifold ,Mbd is obtained by identifying the boundary of V to the boundary of W by the homeomorphism :VW defined by ,,ab cduvuv uv for 1,uvV S  1S. The integers b and d determine ,Mbd , up to ho- meomorphism. An embedded Klein bottle K in ,Mbd is called a Heegaard Klein bottle if for any regular neighborhood NK of K, NK is a twisted I-bundle over K and the closure of  ,MbdN K is a solid torus. Any G-action which leaves a Heegaard Klein bot- tle invariant is said to split. We describe in Section 2, the G-actions (up to equiva- lence) which can act on a prism manifold and the seven possible quotient orbifolds ,i for 17i where  and  are some positive integers. For example, the orbifold 1, is an orbifold whose underlying space is a prism manifold with a simple closed curve as an exceptional set of type .. ,kgcd. The closure of the complement of the exceptional set is a twisted I-bundle over a Klein bottle. Section 3 gives necessary and sufficient conditions for an orbifold of type ,i to be regularly covered by a prism manifold. Let ,i be the partially ordered set of equiva- lence classes of G-actions with orbifold quotient J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 150 ,i. Define a set  1,, :..,1,divides,1 (mod2),andor2.bdgcd bdbddb  We show in Sections 4 and 6 that the set 1, is a distributive lattice which is isomorphic as a partially or- dered set to 1,, and this implies that 1, is also a distributive lattice. For 27i , we show that ,i is isomorphic as a partially ordered set to a union of lattices of type 1,xy. In addition, we give necessary and sufficient conditions for two lattices of type 1,xy to be isomorphic. A G-action is primitive if it does not contain a non- trivial normal subgroup which acts freely. These actions determine minimal elements in the partially ordered sets. We determine the primitive actions for each possible orbifold quotient in Section 5. In Section 6 we compute the maximum length of a chain in the partially ordered sets ,i. Further- more, if 0b is the largest odd divisor of  such that 0.. ,1gcd b and 01ikliibp is the prime decom- position, then we show that 1, is a Boolean al- gebra if and only if 1il for all 1ik. When 1,mn is a prism manifold, we consider in Section 7 a partially ordered set of non-cyclic subgroups ,mn of 11π,mn. We show that ,mn is a lattice isomorphic to 1,mn where the partial order- ing on the groups is given by 21GG if 2G is a sub- group of 1G. The meet 1212,GG GG and the join 1212GGG G. Moreover we show that there exists a sublattice  of ,mn which is a Boolean algebra, and a lattice homomorphism ,mn which re- stricts to the identity on . Section 8 is devoted to several examples which illus- trate some of the main results. 2. Actions on Prism Manifolds In this section we describe a set of G-actions on a prism manifold ,Mbd which leave a Heegaard Klein bottle invariant and their quotient spaces ,Mbd G. We obtain seven quotient types ,i for 17i where  and  are some positive integers. It follows by  that any G-action which leaves a Heegaard Klein bottle invariant is equivalent to one of the actions in Quotient type [i] for some 17i, and ,,iMbd G. By  these actions are completely determined by their restriction to a Heegaard Klein bottle K. We begin by describing G-actions on K and note that these actions extend to all of ,Mbd . We will list the actions by their quotient type. Let Vk be the orbifold solid torus with exceptional set the core of type k and let  BkV k be the Conway ball, where :Vk Vk is the involution defined by ,,uvuv. The Conway sphere Bk has 4 cone points, each of order 2. It is convenient to view the Klein bottle as the set of equivalence classes 1:1 22,,111,, 22.22Kru ruSuuuuu  1) Quotient type 1,. For 21mn, define actions 1:DiffmK and 12:DiffmK by 2π211inru rue and π21111inru uer where 1 represents a generator of the group. The quotients 1K and 1K are both Klein bottles. These actions extend to the prism manifold ,Mbd and we denote these extensions using the same letters to obtain 1:Diff,mMbd and 12:Diff,mMbd. The orbifold quotient for these actions is denoted by 11,Vk W, where 1W is a twisted I-bundle over the Klein bottle and :iVk W is defined by ,,kkuvuvuv where  and  are inte- gers, and ,kgcd . It follows that the quotient 1,Mbd is the orbifold denoted by 121,bn d and the qoutient 1,Mbd is the orbifold denoted by 121,2bn d. 2) Quotient type 2,. For 2mn define ac- tions 2:DiffmK and 22:DiffmK by π21inru rue, and 211, 0ru ur and π21, 0inru rue. The quotients 2K and 2K are both mirrored annuli. These actions extend to all of ,Mbd and we obtain 22:Diff,nMbd and 22 2:Diff,nMbd . The orbifold quotient for these actions is denoted by 22,Vk W, where 2,,, ,1WTIuvtuvt is a twisted I-bundle over the mirrored annulus mA and ψ is defined as in Case 1. The orbifold quotient  22,2,Mbd nbd and 22,2,2Mbdnb d. 3) Quotient type 3,. Define 32 21:n DiffK and 34:DiffnK by 311, 0ru ur, π2131, 0inru rue, and J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 151π2311inru uer. The quotients 3K and 3K are mirrored Möbius bands. These actions extend to the prism manifold ,Mbd and we obtain 3221:n Diff ,Mbd and 34:Diff,nMbd. The or-bofold quotient for these actions is 33,Vk W 3VkW where 3,, ,,1WTIuvtvut  is a twisted I-bundle over the mirrored Möbius band mM. The orbifold quotient  33,21,21Mbdb ndbnd and 33,2,2Mbdnb dnb d. 4) Quotient type 4,. For 21mn, define the action 421:Dih DiffnK by π2141, 0inru rue and 410,1 ru ur. The quotient 4K is the projective plane 22, 2P containing two cone points of order two. This action ex- tends to ,Mbd and we obtain 421:Dih n Diff ,Mbd . The orbifold quotient is 4, 4Bk W where 4,,1WIztzt  is the twisted I-bundle over 22, 2P and  is the homeo- morphism of  induced by . The orbifold quotient  44,(21),Mbdb nd. 5) Quotient type 5,. Define the following ac- tions: 521:Dih DiffnK where π2151,0inru rue and 50,1 ru ru; 542:Dih DiffnK where π21511, 0inru uer and 50,1 ru ru; 542:Dih DiffnK where 2π4251, 0inru rue and 510,1 ru ur; and 54:Dih DiffnK where π251, 0inru rue and 510,1 ru ur. The orbifold quotient for all these actions is the mirrored disk 22, 2. All these actions extend to Diff ,Mbd . If  5,,1WIztrzt , where r is a reflection exchanging a pair of cone points, is the twisted I-bundle over 22, 2, then the orbifold quotient for these ex-tended actions is 55,Bk W. We obtain  55,21,Mbdb nd,  35,21,2Mbdb nd, 55,42,Mbdb nd and 55,4,Mbd nbd. 6) Quotient type 6,. Define actions 66,: 2Dih DiffnK as follows: π611, 0inru uer and 60,1 ru ru if n is even, and 2π611, 0inru uer and 610,1 ru ur if n is odd. The quotients 5K and 5K are both a mirrored disk 22 containig a cone point of order two and two cone points of order two on the mirror. These actions extend to the prism manifold ,Mbd and we obtain 66 2,:Dih n Diff ,Mbd . The orbifold quotient for these actions is denoted by 66,Bk W where 6,,1WIztrzt  the twisted I-bundle over the mirrored disk 22, and r is a reflection leaving two cone points fixed and exchanging the other two cone points. The orbifold quotients 5,Mbd and 6,Mbd are both 6,bnd bnd. 7) Quotient type 7,. Define 72:Dih n DiffK and 722:Dih DiffnK as fol- lows: π71, 0inru rue, 70,1 ru ru, and 711, 0,0ru ur, π70,1,0inru rue, and 70, 0,1ru ru. The quotients 7K and 7K are both a mirrored disk 20 containig four cone points of order two on the mirror. These actions extend to the prism manifold ,Mbd and we obtain 72:DihDiff ,nMbd and 722:DihDiff ,nMbd . If 7,,1WIztrzt , where r is a reflection leaving each cone points fixed, then 7W is a twisted I-bundle over the mirrored disk 20. The orbifold quotient for these extended actions is 6, 6Bk W. We obtain  77,2,Mbd nbd and 77,2,2Mbdnb d. 3. Prism Manifold Covers of Orbifolds In this section we give necessary and sufficient condi- J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 152 tions for when the orbifold ,i, 17i, is cov- ered by a prism manifold. The proofs rely on Section 2. Proposition 1. For the orbifold 1,, there exists a prism manifold cover if and only if either  is odd or 0 (mod 4). Proof. Suppose  is odd. Then there exists a -ac- tion on 1,M such that  11, 1,,MM. If  is even, write 02l where 0 is odd. If  is odd, then there exists a 0-action on 2,lM such that  012, 2,,llMM. Suppose now that  and  are both even where 0 (mod 4). Write 02 and 02m where 0 and 0 are both odd. Then there exists a 02-action on 02,mM such that 00021002,2,2 ,2mm mMM. For the converse, suppose that  and  are both even and there is a covering ,,iMbd. Then ei- ther 21bn and d, or 21bn and 2d. Since  is even, it follows that 2 divides b. In the first case, 2 would also divide d, contradicting the fact that b and d are relatively prime. If 0 (mod 4), then again 2 divides d giving a contradiction. Proposition 2. For the orbifold 2,, there exists a prism manifold cover if and only if 0 (mod 2). Proof. Suppose that 0 (mod 2). Write 02. Then there exists a 02-action on 1,M such that  0221, 1,,MM. For the converse, suppose that 2,,Mbd. Then either 2nb and d, or 2nb and 2d. Proposition 3. For the orbifold 3,, there exists a prism manifold cover if and only if  (mod 2). Proof. Suppose that  (mod 2) and let 2d. Suppose that 0 (mod 4), and thus there exists an integer n such that 4n . There exists a 4n- action on 1,Md such that  4331,1,2, 2,.nMdMd ndnd If 0 (mod 4), then write 212n for some n. There exists a 22 1n-action on 1,Md such that 322 131,1,21,21,.nMdMdn dn d  For the converse, suppose that 3,,Mbd. Then either 21bnd and 21bnd, or 2nb d and 2nb d for some n. Subtracting the two equations in both cases, we obtain 2d. Propo sitio n 4. For the orbifold 4,, there exists a prism manifold cover if and only if either  is odd or  is odd. Proof. Since 1,Mbd always double covers 4,Mbd, using a proof similar to that in Proposition 1 shows that there is a prism manifold covering of 4, if and only if  or  is odd by . Proposition 5. A prism manifold covering for the orbi- fold 5, always exists. Proof. Suppose  is an odd number. Then 1,M admits a Dih-action whose quotient is 5,. If β is even, we write 02m where m ≥ 1, 02n where 0n, and 0 and 0 are both odd numbers. If n = 0 or 1n, then 2,mM and 02,mM admit 0Dih and 02Dih-actions respectively, whose quotient space is 5,. If n and m are both greater than 1, or if 1m and 2n, then 1,M admits a 1042Dih m or a 02Dih-action respectively, whose quotient space is 5,. Proposition 6. For the orbifold 6,, there exists a prism manifold cover if and only if  (mod 2). Proof. Since 3,Mbd double covers 6,Mbd and 3,Mbd double covers 6,Mbd, the re- sult follows by Proposition 3. Proposition 7. For the orbifold 7,, there exists a prism manifold cover if and only if 0 (mod 2). Proof. Since 2,Mbd double covers 7,Mbd and 2,Mbd double covers 7,Mbd, the re- sult follows by Proposition 2. 4. Poset of Actions on Prism Manifolds Recall that two group actions :DiffGM and :DiffGM are equivalent if there is a homeo- morphism :hM M such that 1:Gh Gh. If :DiffGM is an action, let :MM be the orbifold covering map. Let  be the set of equivalence classes of actions on prisim manifolds which leave a Heegaard Klein bottle invariant. Now  is partially ordered by setting  if there is a covering :MM such that . Note that the covering :MM is also a regular covering. For a pair of positive integers  and  let 1, denote the equivalence classes of those actions whose quotient type is 1,. Note that by Proposition 1 the set 1, is nonempty if and only if either  is odd, or 0 (mod 4). Unless otherwise stated, we assume from now on that  and  are integers where either  J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 153is odd or 0 (mod 4). Let  1,, :..,1,divides,1 (2)and,or2.bdZZgcd bdbmodddb  It follows that 1, is a partially ordered set under the ordering 22 11,,bdbd if 21bb and 21dd. Let 1, be the subset of 1, consisting of all ordered pairs  1,,bd where d. Note that  11,, if  is odd. Moreover, if 02 (mod 2) and β is even, then 11,,2 . Proposition 8. Let 11,bd and 22,bd be elements of the poset 1,. There exists elements ,bd and ,bd in 1,, such that  11,,bdbd and  22,,bdb d, and 11,,bd bd and ,bd 22,bd . Proof. Let 12.. ,b gcdbb. Note that since id or 2 for 1,2i, it follows that if 12min ,ddd, then idd. Thus b divides  and d is  or 2. If b is even, then it follows that 2 divides both 1bb and 2bb, contradicting 12.. ,bgcdbb. Thus b is odd showing  1,,bd. Moeover 11,,bdb d and  22,,bdb d. Let 12.. ,blcmbb and d 12max ,dd. It follows that b is odd, and hence  1,,bd. Furthermore 11,,bd bd and  22,,bdb d. Corollary 9. 1, is a lattice where for 11,bd and 22,bd in 1, the join 112 21212,,..,,min,bdb dgcdbbdd , and the meet 112 21212,,..,,max,bdb dlcmbbd d . Furthermore, 1, is a sublattice of 1,. Proposition 10. Let 11,bd and 22,bd be elements of 1, such that 22 11,,bdbd. Then there exists either a standard m-action 1 on 22,Mbd , or a standard 2m-action 1 on 22,Mbd , which we denote by , and a regular covering  22 2211:, ,,Mbd MbdMbd. Proof. If 22 11,,bdbd, then 21bb and 21dd. Furthermore 1d, or 12d and 2d, or 22d. Now 12bmb, 1121bn, and 2121bn for some integers m, n1 and n2. Since 112221 21bnb n , it follows that 2121 21nmn , and therefore m must be odd. Since 21dd, the only possibilities are 12dd or 212dd. If 12dd, then there exists a m-action 1 on 22,Mbd such that 2222 111,, ,Mbd MbdMbd. If 212dd, then there exists a 2m-action 1 on 22,Mbd such that 2222 111,,,Mbd MbdMbd. Proposition 11. Let ,bd be an element of 1,. Then there exists either a standard 21n-action 1 on ,Mbd , or a standard 22 1n-action 1 on ,Mbd which we denote by , and a regular covering 1:, ,,Mbd Mbd. Proof. Write 21nb. If d, then there is a 21n-action 1 such that 11,, ,Mbd Mbd. If 2d, then there is a 22 1n-action 1 on ,Mbd such that 11,, ,Mbd Mbd. Theorem 12. For each pair of positive integers  and , the poset 1, is isomorphic to the poset 1,. Proof. Define a function 11:, ,f as follows: let 1,,bd. There exists either a stan- dard b-action if d, or a standard 2b-action if 2d on ,Mbd , which we denote by , such that 1,, ,Mbd Mbd. Define 1,,fbd. Suppose 1112 22,,fbd fbd . Since 1 and 2 are equivalent, there exists a homeomorphism 1122:, ,hMbdMbd such that 112Gh Gh. Since 11,Mbd and 22,Mbd are homeomorphic, it follows that 12bb and 12dd, showing f is one-to-one. Let 1,. Then there exist a prism manifold ,Mbd such that 1:, ,,Mbd Mbd. We may assume that b and d are both positive. By , η is equivalent to one of the standard actions 1 or 1, and 1,21,Mbdb nd or 121,2bn d respectively, for some positive integer n. Therefore 1b (mod 2) and d or 2d. If  is either 1 or 1, then ,fbd, showing f is onto. Suppose now that  22 11,,bdbd. Let 111,fbd and 22 2,fbd where 1 and 2 are the standard J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 154 11b and 22b-actions respectively on 11,Mbd and 22,Mbd and 1i or 2. We have the coverings  1111111:, ,,MbdMbd and  11111 11:, ,,Mbd Mbd. By Proposition 10 there is a standard 12bb-action  on 22,Mbd where 1 or 2, and a regular covering covering  22 2211:, ,,Mbd MbdMbd. Since these are standard actions and 212 1bbb b it fol- lows that 21. This shows that 21. Corollary 13. 1, is a lattice. We will now consider maximal and minimal elements in 1,. Write 02n where 0 is odd. Then the maximal element in 1, is 2,n if  is odd, and 2,2n if  is even. Note that if ,bd 1,, then 2n divides b and 2nb is odd. In de- scribing the minimal elements let 0b be the largest odd divisor of 0 such that 0.. ,1gcdb. If  is odd or if 0n, then the minimal element in 1, is 02,nb, otherwise the minimal element is 02,2nb. We say an element 11,bd is directly below 22,bd or that 22,bd is directly above 11,bd if whenever  112 2,,,bdbdb d, then either 11,,bd bd or  22,,bdb d. Theorem 14. Let 00112,mb and 00222,nc be the minimal elements in 111, and 122, respectively where 1i or 2, and let 01ikmiibp and 01isniicq be the prime decompositions. Sup- pose one of the following holds: 1) 1 and 2 are both odd. 2) 1 and 2 are both even and 00 0mn. 3) 1 and 2 are both even and 000mn. 4) 1 even with 00m and 2 odd. Then 111, is isomorphic to 122, if and only if ks and after reordering iimn for 1, ,ik. If 1 is odd and 2 is even with 00n, then 111, is isomorphic to 122, if and only if 1ks, after reordering iimn for 1, 2,,1ik, and 1km. Proof. We will first assume that 1 and 2 are both odd. Suppose 11112 2:, ,f is an iso- morphism. Now 012,m and 022,n are the maxi- mal elements of 111, and 122, respec- tively, and 00122, 2,mnf. The elements directly below 012,m in 111, are 0112,,,mp 012,mkp and the elements directly below 022,n in 122, are  0012 22,,,2,nnsqq. Since f must take the elements directly below 012,m to the elements directly below 022,n, it follows that ks. The elements directly above 01,b in 111, are listed as 0012011112 112112,,2 ,,,2,iikimmm mmmiiiimmmkiikpppppp Similarly the elements directly above 02,c are 0012011122 212122,,2 ,,,2,.iikinnnnnniiiinn nkiikqqq qqq By reordering we may assume  0011122,2,jjiimnmmnnji jiij ijfppq q for 1jk. The number of elements in 111, is 11kiim, and this equals the number of elements in 122, which is 11kiin. Let 01111 111111,, ,:2,,.jijmmmjiijbppb Similarly let 01122 222122,, ,:2,,.jijnnnjiijcqqc  It follows that 11112 2,,jjf. Thus the number of elements in 111,j which is 1jiijmm is equal to 1jiijnn the num- ber of elements in 122,j. Using the equations  111kkiiij imn and 11jijiij ijmmnn, we obtain 11,11ijij jjijijnmmnmn and this implies that jjmn for 1jk. We now suppose that 0012,mb and 0022,nc are the minimal elements in 111, and 122, respectively, and 01ikmiibp and 01ikmiicq are the prime decompositions. If   1111,,b, then J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 155012ikmsiibp where 0iism . Define   11112 2:, ,f by 0121,2 ,iknsiifb q. It is not hard to check that f is an isomorphism. The proof in cases (2) - (4) is similar. We now assume that 1 is odd and 2 is even with n0 = 0. Since the argument is similar to the previous case, we will sketch the proof. The elements directly below 012,m in 111, are  0011 12,,,2 ,mmkpp and the elements directly below 21, 2 in 122, are 122 2,2,,,2,1,sqq. It follows that k = s + 1. The elements in 111, directly above 0012,mb are  0012011112 112112,,2 ,,,2,,iikimmm mmmiiiimmmkiikpppppp and the elements directly above 02,c are  121111222121121,, ,,,,,iikinnnniiiinnkiikqqq qqq 02,2c. By relabeling we may assume that 011122, ,jjiimnmm nji jiij ijfppq q for 11jk and 011022,,2kimm mkiikfppc. Now  11112 1kkiiiimn , and for 1jk we have  12 1ji jiij ijmm nn . Using these two equations we obtain jjmn for 11jk. The number of elements greater than or equal to 0112,kimm mkiikpp and 02,2c is 1kiikmm and 111kiin respectively. Since these two numbers must be equal, it follows that 1km. For the converse suppose that 01ikmiibp where 1km and 101ikmiicq. If 1,b is any element in 111,, then 012ikmsiibp where 0iism , and 0ks or 1. Let 11121,,2ksiifb q if 0ks, and 112=1 ,ksiiq if 1ks. It follows that f is an isomorphism. For a pair of positive integers  and , let 2, denote the equivalence classes of those actions whose quotient type is 2,. Let  2,, :..,1,divides,0 (2),andor2.bdgcd bdbmodddb It follows that 2, is a partially ordered set under the ordering 22 11,,bdbd if 21bb, 121bb (mod 2), and 21dd. The proof of the following theorem is similar to that of Theorem 12. Theorem 15. For each pair of positive integers  and , the poset 2, is isomorphic to the poset 2,. We will now consider the structure of the partially or-dered set 2,. Write 012mbb where 0b and 1b are both odd and 0b is the largest odd divisor of  which is relatively prime to . Theorem 16. For each pair of positive integers  and , the poset 2, is a disjoint union of lat- tices given by 21101011110 0,2, if0(2),if0(4)(2,2),if0 (2)2mmjjmmjjbmodbmodbb mod Proof. We first assume that  is odd. Note that for 1jm, we have 202, ,mjb. It suffices to show that if 02, ,mjbbd, then 02,mjb ,bd and hence 02,mjb is a minimal element, and if 2,,bd, then 0,2,mjbd b for some unique j where 1jm. Suppose 02,mjb ,bd , and thus 02mjb divides b,  divides d, and 02mjbb is odd. Now d divides , which implies d. Since b divides 012mbb and .. ,1gcdb, it fol- lows that b must divide 02mb. Write 2wbb where b is odd. Now 022wmjbb being odd implies wmj and b0 divides b. Note that 2mjbb divides 02mb. Thus b divides 0b showing that 0bb, and there-fore 02mjbb. Let 2,,bd. Since  is odd, it follows that d. As above we have that b divides 02mb. Furthermore, 02mbb must be even. We may write 2rbb where 01rm, b is odd, and b di- vides 0b. Therefore 0,2,rbd b. We now assume that  is even and we write 2n where  is odd. There are two cases to consider: n ≥ 2 and n = 1. Suppose first that n ≥ 2. Now 0,b 2,. We will show that 0,b is the minimal element in 2,. Suppose that 2,,bd and 0,,bbd. In this case d. Also since 0bb J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 156 and 0b are both odd, it follows that b is odd. Since .. ,1gcdb and b is odd, it follows that b divides 0b. Thus 0bb showing that 0,b is the minimal ele- ment. Now let  2,,bd. Recall that d or 2d. If d, then since b and d are relatively prime, b must be odd. Furthermore, b must divide 0b, and thus 0,,bd b. If 2d, then 122nd and d is even since n ≥ 2. Now b and 12nd are relatively prime, which implies that b is odd. Again we have that b must divide 0b, so that 0,,bd b. This shows that 0,b is the minimal element and 210,,b. We now consider the case when n = 1. Note that 202,2,mjb for 1 ≤ j ≤ m – 1 and 20,,b. We need to show that these are the minimal elements in 2,, and if ,bd 2, then either 0,2,2mjbd b for some unique j or 0,,bd b. The proof to this is similar to case 1. Remark 17. Note that by Theorem 7 each 1102,mmjjb and 11102,2mmjjb is a dis- joint union of isomorphic lattices. For a pair of positive integers  and  with > and  even, let 3, denote the equivalence classes of those actions whose quotient type is 3,. Let 3,,: ,2. .,1,divides.2bd dgcdb db  It follows that 3, is a partially ordered set under the ordering 21,,bdbd if 21bb and 121bb (mod 2). Theorem 18. For each pair of positive integers  and  with > and  even, the poset 3, is isomorphic to the poset 3,. Proof. Let  3,,bd and let 2mb. Ob- serve that bmd and bm d. There exists a standard 2m-action 2:Diff,mMbd such that 33,,,Mbdbmdbm d where 3 if m is odd and 3 if m is even. Define  33:, ,f by 3,,fbd. Suppose that 21,,bd bd. Let 112mb and 222mb and let 112 1:Diff,mMbd and 222 2:Diff,mMbd be the standard actions. It follows that 1212bmmb, and therefore 12m is a sub- group of 22m. Furthermore, 122111bb, which implies that 21 and thus f is order preserving. The proof that f is one-to-one and onto is similar to that in Theorem 12. We will now consider the structure of the partially or- dered set 3,. Write 22m where  is odd. Let 0b be the largest positive odd divisor of  which is relatively prime to 2d. Theorem 19. For each pair of positive integers  and  with >,  even, and 2d, the poset 3, is a disjoint union of isomorphic lat-tices given by 1003102,if 0(4),,if0(4)mmjjbd modbd mod  Proof. Suppose first that 0 (mod 4) (equiva- lently d is odd). Note that 02jb divides 2 for 0jm, and since d is odd we have 0.. 2,1jgcdbd. It follows that 302, ,jbd and 02,jbd is a minimal element of 102,jbd. Let 3,,bd. Write 2kbb where b is odd. Since b divides 22m, it follows that 0km and b divides . Furthermore, b and d are relatively prime. Since 0b is the largest positive odd divisor of  which is relatively prime to d, it follows that b divides 0b. Hence 10,2,kbdb d for a unique k. Now sup- pose 10,2,kbdb d for some k. By assumption 2d. Since b divides 02kb and 02kb divides 22m, it follows that b divides 2, and hence 3,,bd. The proof for 0 (mod 4) is similar. Corollary 20. 1003102,if 0(4),,if0(4)mmjjbd modbd mod  For each pair of positive integers  and , let 4, denote the set of equivalence classes of actions J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 157on prisim manifolds whose quotient space is 4,. Theorem 21.  41,, Proof. If 4,, then  is equivalent to 521:DihDiff,nMbd for some integers n, b, and d where 21bn and d. Since ,bd 1,, define a function 41:, ,f by ,fbd. It follows easily that f is an order preserving surjection. For each pair of positive integers  and , let 5, denote the set of equivalence classes of actions on prisim manifolds whose quotient space is 5,. We now consider the structure of the partially ordered set 5,. Write 012mbb where 1b is odd and 0b is the largest odd divisor of  that is relatively prime to . Theorem 22. 51001011010,2,if0(2)and1,if0(4)and12,2,if0(2)and 12,if1mjjmbmodmbmodmbmodmbm  Proof. Suppose that 0 (mod 2). Let 5,. We have a covering 5,,,Mbd Mbd for some positive integers b and d. Now  is equivalent to either of the standard actions 5, 5 or 5. The action 5 is impossible since  is odd. We will de- fine a function 5100:, 2,mjjfb as fol- lows: if  is equivalent to 5, then 01212mbn bb for some n and d. Since b and d are relatively prime and 0b is the largest odd divisor of  that is relatively prime to , it follows that b divides 02mb. Thus 10,2,mbd b and we let 10,2,mfbdb. If  is equivalent to 5, then 01422mbn bb , and this implies that b must divide 102mb. Thus ,bd 1102,mb and we define 110,2,mfbdb. If  is equivalent to 5, then 0142mnbb b for some n. Write 02knn where 0n is odd. This implies that b divides 202mkb where 02km . This shows that 120,2,mkbd b and we define 120,2,mkfbdb. We now show that f is an order preserving bijection. Note that there do not exist integers n, n, b, and b, such that 21221bn bn , or 214bn nb , or 22 14bn nb , if either b divides b or b di- vides b with odd quotient. This implies that f is one-to- one. Furthermore if 21, then 2 and 1 are both equivalent to either 5, 5 or 5. From this it can be shown that 21ff. To show f is onto, suppose 10,2,jbd b for 0jm. Let 0221jbnb for some positive integer n. If mj, then 01 1221mbbbnb, and since 1b is odd we may write 21bn. Hence there is an action 521:DihDiff ,nMbd such that 55,,Mbd, and thus 5f ,bd . Similarly, if 1jm or if 1jm, we ob- tain actions 5 and 5-actions respectively. This shows that f is onto. If 0 (mod 4), then if m = 1 there exist only 5- actions, and if m > 1 there exist only 5-actions. For 02 (mod 2), if m = 1 there exist only 5 and 5- actions, and if m > 1 there exist only 5 and 5-actions. If  is odd and  is even, then there exist only 5 and 5-actions. The proof in all these cases is similar to the above. For each pair of positive integers  and  with > and  even, let 6, denote the set of equivalence classes of actions on prisim manifolds whose quotient space is 6,. Theorem 23. For each pair of positive integers  and , the poset 6, is isomorphic to the poset 3,. Proof. Let 6,1. Now  is a Dih n- action on a prisim manifold ,Mbd and is equivalent to 6 if n is even or 6 if n is odd. Furthermore, 6,,Mbdbn dbn d, and therefore bn d and bn d. It follows that 3,,bd. Define a function 6:,f 3, by ,fbd. Let 3,,bd. Therefore 2bn and 2d for some n. This implies bnd and bn d. If n is even there exists an 6-action, and if n is odd there exists an 6-action. Therefore f is onto. Let 111,fbd and 222,fbd and suppose 12ff. Then  112 2,,bd bd and hence 12bb and 12dd. We also have 11 1bnd and 22 2bn d so that 12nn. Recall that 1 and 2 are equivalent to either an 6 or 6-action. If 12nn is even, then both of them are equivalent to an 6-action, otherwise they are both equivalent to a 6- J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 158 action. Hence 1 and 2 are equivalent showing f is one-to-one. If 12, then there is a covering map  22 11:, ,Mbd Mbd where 12dd. Hence, 2121nbb for some n which shows 21bb is an odd number. Therefore, we conclude  1122,,bdb d showing f is order preserv- ing. For each pair of positive integers  and , let 7, denote the set of equivalence classes of ac- tions on prisim manifolds whose quotient space is 7,. The proof of the following theorem is similar to that of Theorem 23. Theorem 24. For each pair of positive integers  and , the poset 7, is isomorphic to the poset 2,. 5. Primitive Actions on Prism Manifolds Let :Diff ,GMbd be a G-action on a prism manifold ,Mbd with orbifold covering map  :, ,Mbd Mbd. We say that  is primitive if G does not contain a non- trivial normal subgroup which acts freely on ,Mbd . Therefore for any nontrivial normal subgroup H of G, if 0:Diff ,HHMbd , then 0,Mbd is not a manifold. In this section we determine when an action is primitive. Theorem 25. Let :Diff,mMbd be a m- action on the prism manifold ,Mbd . 1) If  is equivalent to 1, then  is primitive if and only if for every prime divisor p of m, 0d (mod p). 2) If  is equivalent to 1, then  is primitive if and only if b is even and for every odd prime divisor p of m, 0d (mod p). 3) If  is equivalent to 2 or 3, then  is primi- tive if and only if either 2nm, or if p is any odd prime divisor of m, then 0d (mod p). 4) If  is equivalent to 3, then  is primitive if and only if either 2m, or if p is any odd prime divisor of m, then 0d (mod p). Proof. We may suppose 1. Then 21mn and if l is a subgroup of m and 0:ll Diff ,Mbd , then  10,,,Mbdbld Mbd. Furthermore 1,bl d is a manifold if and only if .. ,1gcdbl d. Assume that  is primitive and let p be a prime divisor of m. Consider the subgroup p. Since .. ,1gcdb d and  is primitive, it follows that  ..,..,gcdbp dgcdpdp. Thus p divides d. Now suppose that every prime divisor of m also divides d and let l be a subgroup of m. Let p be a prime divisor of l. Since l divides m, it follows that p divides m. Hence by assumption .. ,1gcdbl d, showing that  is primitive. For part 2), suppose that 1. Then 22 1mn and if l is a subgroup of m and 0:ll Diff ,Mbd , then either 21ls and 10,,,Mbdbld Mbd, or 22 1ls and 10,21,2,Mbdb sdMbd. Furthermore 1,bl d is a manifold if and only if .. ,1gcdbl d; and 121,2bs d is a manifold if and only if ..21,21gcdbsd. Assume first that  is primitive. If p is an odd prime divisor of m, then the same argument used in the 1 case shows p divides d. Now 2 is a subgroup of m and we have a covering 12,,2,MbdbdMbd. Since  is primitive, 1,2bd is not a manifold, and since .. ,1gcdb d it follows that 2 divides b. For the converse suppose that b is even, and if p is any odd prime divisor of m, then 0d (mod p). Let l be any sub- group of m. If l is odd, the proof that  is primitive is identical to the 1 case. If l is even, then ..21,21gcdbsd, showing that  is primitive. To prove part 3), suppose that  is equivalent to 2. Therefore 2ms and 22:Diff,sMbd is defined by π21isru rue where ru is any point in the Heegaard Klein bottle K and 1 denotes a generator of 2s. If l is any subgroup of 2s and 2:Diff,llMbd, then 2π1ilru rue where 1 denotes a generator of l. We now suppose that 2 is primitive and p be an odd prime divisor of 2ms. Letting lp, we obtain a covering 1,, ,MbdMbdbpd. Since 2 is primitive .. , 1gcdbp d, and since ..,..,gcdbp dgcdpd, it follows that p divides d. Suppose 2nm. Then 2tl and 2ππ1221iittru ruerue    . Now K is a mir- rored annulus, showing that ,Mbd is not a mani- fold, and thus 2 is primitive. Suppose now that 2nm, and if p is an odd prime divisor of m, then 0d (mod p). Assume there is a covering 1,, ,MbdMbdbld (Note that the quo- tient space cannot be 1,2bl d by definition of 2 J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 159and 1). It follows that l is odd. Let p be any odd prime divisor of l. It follows that p divides m, and hence by assumption p divides d. Thus .. ,1gcdbl d, showing 2 is primitive. We now suppose that 3. In this case 4ms and 34:Diff,sMbd where π2311isru uer. If l is a subgroup of m, let 3:Diff,llMbd. Assume first that  is primitive. Suppose p is an odd prime divisor of m and consider the primitive subgroup p of m. Here lp, and since mp is even we have 2π1ipru rue. Since  is a primitive 1-action on ,Mbd , it fol- lows by the above that 0d (mod p). Now suppose 2nm and let l be a subgroup of m. Then 2tl and π121itru rue. In this case  is an 2- action and ,Mbd is not a manifold. Now suppose that every odd prime divisor of m also divides d. Let l be a subgroup of m. If l is odd, then 4sl is even and 2π1ilru rue which is an 1- action. The result follows by the above that  is primi- tive, and thus  is primitive. Now suppose l is even. If ml is even then 2ππ1iillru ruerue   , which is an 2-action; and if ml is odd then 2ππ111iillru ueuerr, which is a 3-action. In either case the quotient is not a manifold. Hence  is primitive. We now prove part 4) and suppose 3. Therefore 22 1mn and 322 1:Diff,nMbd where 2π21311inru uer. If 2m, then 3K is a mirrored Möbius band and the action is primitive. So suppose that 2m. Assume first that 3 is primitive and let p be an odd divisor of m, and hence p divides 21n. Consider the subgroup p and let 3:Diff,ppMbd. Then 4π1ipru rue, and we have a covering  1,, ,MbdMbdbpd. Again as above since 3 is primitive, p must divide d. We now suppose that for each odd prime divisor p of m, 0d (mod p). Let l be a subgroup of m. If l is odd we obtain a covering 1,, ,lMbdMbdbld, and as above if p is a prime divisor of l we obtain .. ,1gcdbl d. Thus the action is primitive. If l is even then 22 1ls, and if 3l then 2π2111isru uer. In this case 3,,lMbd bld, which is not a manifold show- ing that the action is primitive. Proposition 26. Let 22:Diff,mMbd be an action on the prism manifold ,Mbd where 2m. Then  is primitive if and only if either 2nm, or if p is any odd prime divisor of m, then 0d (mod p). Proof. We may assume that 2, and therefore 222m. Suppose that 2 is primitive. This implies that 2 is primitive and the result follows from Theo-rem 25. Now suppose that either 2nm, or if p is an odd prime divisor of m, then 0d (mod p). Note that this implies by Theorem 25 that 2 is primitive. Let H be a subgroup of 22m and 2H. If 211HZ, then ,Mbd is not a manifold. So we may assume that 211HZ , and hence H is a subgroup of 2m. Since 2 is primitive, ,Mbd is not a manifold showing that 2 is primitive. Proposition 27. Let 22:Diff,Mbd be an action on the prism manifold ,Mbd . Then  is primitive if  is equivalent to either 5, 6 or 7. If  is equivalent to 2 or 5, then  is primitive if and only if 0b (mod 2). Proof. If  is either 5, 6, 7, 2 or 5, then any subgroup of 22 restricted to a Heegaard Klein bottle K is a product of the following homeomorphisms where ru K: πiru rue, ru ru, and 1ru ur. The only fixed-point free action on K is the homeomorphism π1iru uer. By definition of 5, 6 or 7, they do not contain this homeomor- phism, and hence they are primitive. But the actions 2 and 5 do contain this 2Z subgroup and we obtain a covering 21,, ,2.MbdMbdbd Since 1,2bd is a manifold if and only if 2 does not divides b, the result follows. Proposition 28. Let :DihDiff ,mMbd be an action on the prism manifold ,Mbd for m > 2. 1) If  is equivalent to either 4 or 5, then  is primitive if and only if for every prime divisor p of m, 0b (mod p). 2) If  is equivalent to 5, then  is primitive if and only if b is even and for every odd prime divisor p of J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 160 m, 0d (mod p). 3) If  is equivalent to either 5, 5, 6 or 7, then  is primitive if and only if either 2nm, or if p is an odd prime divisor of m, then 0d (mod p). 4) If  is equivalent to 6, then  is primitive if and only if either 2m or, if for each odd prime divi- sor p of m, 0d (mod p). Proof. We may assume  is either 4 or 5, and thus 21mn and 1|m. If  is primitive, then 1|m is primitive, and the result follows by Theo- rem 25. For the converse, suppose that H is a subgroup of 12Dihmm. If 21H, then ,Mbd H is not a manifold. So we may assume 21H, and thus H is a subgroup of m. Hence lH for some l with l dividing m. By Theorem 25, 1 is primitive, and so ,Mbd H is not a manifold. Thus 4 and 5 are primitive. The proof for the other cases is similar to Theorem 25. Proposition 29. Let 22:DihDiff,mMbd be an action on the prism manifold ,Mbd . 1) If 2m, then  is primitive if and only if either 2nm, or if p is any odd prime divisor of m, then 0d (mod p). 2) If 1m, then  is primitive if and only if 0b (mod 2). Proof. We may assume that 7. Since 2272m , using Propositions 26 and 27, a proof similar to that used in Proposition 28 proves the result. 6. Lattice Structure In this section we compute the maximum length of a chain in the partially ordered sets ,i. In addition, we give necessary and sufficient conditions for 1, to be a Boolean algebra. Theorem 30. Let 11,bd and 22,bd be elements of 1,, and so iid, where i is 1 or 2 for 1, 2i. Suppose 22 11,,bdbd. Then 2b divides 1b and 21. 1) If 21, then there exists ,bd in 1, such that 22 11,,,bdbdbd. 2) If 12, then 12bb is prime if and only if there does not exist ,bd in 1, such that 22,bd  11,,bdb d. Proof. Since 11,bd and 22,bd are elements of 1, for 1, 2i, it follows that .. ,1iigcd b d, ib divides , 1ib (mod 2), and iid where 1i or 2. By the definition of 22 11,,bd bd, we have 21bb and 21dd. Now 1221dd, which implies 21. Suppose 21. Thus 22 and 11, hence 212dd. Now 112,,bd and 22 1211,>,>,bd bd bd showing (1). We now suppose that 12, and thus 12dd. Sup- pose there exists ,bd in 1, such that 22 11,,,bdbdbd. It follows that 12ddd, 2bb and 1bb. Therefore 1122bbbbb b . Note that 12bb is prime if and only if either ,bd equals 22,bd or 11,bd . Corollary 31. Let 1 and 2 be elements of 1,, such that 111:Diff,mMbd and 22222:Diff,mMbd where mi is odd and i is either 1 or 2. Suppose 21. Then 2b divides 1b and 21. 1) If 22 and 11, then there exists 1, such that 21 . 2) If 12 and 12bb is prime, there exists no 1, such that 21 . Recall that the maximal element in 1, is 2,n if  is odd, and 2, 2n if  is even. To obtain the minimal element let 0b be the largest odd divisor of  such that 0.. ,1gcdb. The minimal element is 02,nb if either  is odd or if 0n, otherwise the minimal element is 02,2nb. Theorem 32. For the partially ordered sets ,i where 17i and 3,6i, let 12122knnnnkpp p and 12122kmmmmkpp p be the prime decompositions. Let 0b be the largest odd divisor of  relatively prime to . Thus, 12012klllkbpp p where 0jl if and only if ,0jjmin nm and jjln if and only if ,0jjmin nm. Then the following chart gives the length of a maximum chain in each ,i. The maximum length of a chain Ordered set Conditions Max. length1,, 2,, 7, 0 (mod 2) 11kiil1,, 2,, 7, 0 (mod 2) 12kiil4, none 11kiil5, 0 (mod 2) and n = 0 or n ≥ 1 11kiil5, 0 (mod 2) and n = 0 12kiil J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 161Proof. We will consider first 1,. Since 1, is isomorphic to 1, we will prove the result for 1,. Suppose that β and  are both even, and thus 0,2b and 2,2n are the minimal and maximal elements of 1, respectively. We may construct a chain in 1, 1200101101,2< ,2<,2<<,2< <2,2<2,2lnnkbbpbpbp p such that dividing any two consecutive first coordinates yields a single prime in the prime decomposition of 0b. Now the length of this chain is 11kiil. Since any maximal chain must contain both the minimal and maximal elements, it follows by the above theorem that this is a maximal chain. The other cases for 1, are similar. For the case 2, note that 2, is iso- morphic to 2,, which by Theorem 16 is equal to 1102,mmjjb if 0 (mod 2); 10,b if 0 (mod 4); or 11 110 02,2 ,mmjjbb if 02 (mod 2). By Remark 17 each 1102,mmjjb and 11102,2mmjjb is a disjoint union of isomorphic lattices. The result now follows by applying the above 1, case. By Theorem 21,  41,,. If  is odd then 11,,, and the result follows by the above. If  is even, then a chain in 1, can be con- structed as the above where the second coordinate is re- placed by  proving this case also. By Theorem 22, 5, is isomorphic to 1002,mjjb if 0 (mod 2) and m ≥ 1; 10,b if 0 (mod 4) and m ≥ 1; 1102,2 ,mb if 02 (mod 2) and m ≥ 1; or 10,b if 0m. Using the above results, the first three cases give the maximum length of 11kiil. When 0m, then there is no restriction on . Thus the maximum length is 11kiil if  is odd and 12kiil if  is even. For the remaining case, it follows by Theorem 24 that  72,,, which in turn is isomorphic to 2,. The result now follows by the above. Theorem 33. For a pair of positive integers  and  with >,  even, let 22m where  is odd. Let 0b be the largest odd divisor of  rela- tively prime to 2d and let 12122ksssskppp and 12122kttttkpp p be their prime decompositions. Thus, 12012klllkbpp p where 0jl if and only if ,0jjmin st and jjls if and only if ,0jjmin st. For the partially ordered sets 3, and 6, the maximum length of a chain is 11kiil. Proof. By Theorems 18 and 23 it follows that 3, and 6, are both isomorphic to 3,, which is equal to 1102,mmjjbd if 0 (mod 4) or 1, if 0 (mod 4). In both cases the maxi- mum length is 11kiil. A lattice L is a distributive lattice if for any a, b, c in L,  abcab ac. Proposition 34. 1, is a distributive lattice where for 11,bd and 22,bd in 1, the join 112 21212,,..,,min,bdb dgcdbbdd and the meet 112 21212,,..,,max,bdb dlcmbbd d . Proof. By Corollary 9, 1, is a lattice. A com- putation using the following equation max ,min,minmax ,,max ,lmnlm ln for any positive integers l, m and n, shows that 1, is a distributive lattice. Remark 35. If we represent the minimal element by 02,nb and the maximal element by 2,n where  is either 1 or 2, then for any element 12, ,nbd we have  02,2,2,nn nbd bbd and  2, 2,2,nn nbd bd. A lattice ,,B is said to be a Boolean algebra if the following hold: 1) B is a distributive lattice having a minimal element 0 and a maximal element 1. 2) For every aB, 0aa and 1aa . 3) For every aB there exists aB such that 1aa and 0aa. Proposition 36. For the partially ordered set 1, let 0b be the largest odd divisor of  such that 0.. ,1gcdb and let 01ikliibp be the prime decomposition. Then 1,,, is a Boolean algebra if and only if 1il for all 1ik. Proof. Suppose that 1il for all 1ik. By Remark 35 above and Proposition 34, it remains to show (3) of the definition. Let 1112, ,nbd. Now 1022nnbb J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 162 and so let 021bbb. Observe that 12.. ,1gcd bb and 1212 0.. ,lcmbbbbb. If either 0n and  is even, or if  is odd, then all the elements in 1, have the same second coordinate—either 2 in the first case or  in the second case. In this case 11 21012,2,2 ,nnnbd bdbd and 11 2112,2,2,nnnbdb dd. The remaining case is 0n and  even. If 1d then let 22d, and if 12d we let 2d. It follows that  112 22, 2,nnbdb d gives the minimal element and  112 22, 2,nnbd bd gives the maximal element. We now suppose that 1,,, is a Boolean algebra. Suppose there exists an >1jl. The minimal element in 1, is 02,nb where  is either 1 or 2. Now 102,njbp is an element of 1, and 1002,>2,nnjbp b. There exists a com- plement 2,nbd such that 1002,2,2,nn njbd bpb, and so 100.. ,jlcmbpbb. It follows that pj divides b. We also have  102, 2nnjbd bp equal to the maximal element 2,n, and so 10..2 ,22nn njgcdbb p. But since >1jl, it follows that pj divides 10jbp, giving a contradiction. Proposition 37. Let 00,Ibd be an ideal of a lattice 1, such that 00,bd is directly below 2,nd which denotes the maximum element in the lattice. If 112 2,,bdb dI, then 11,bd I or 22,bd I and I is a maximal ideal. Proof. Let 112 20 0,, ,bdb dIb d. Suppose both ,iibd I for 1, 2i. Since there is no element between 00,bd and 2,nd, we have  00,,2,niibdb dd, where 0,2nig.c.d bb for 1, 2i. This says that ib and 0b do not have a com- mon odd prime divisor for 1, 2i. On the other hand, 112 20 0,, ,bd bdbd so that 012.. ,blcmbb . Since 0b and ib do not have any common odd prime divisors, this forces 02nb. As 00 0,2,nbd d is not the maximum element, 02dd. This result is possible only when the second coordinate is allowed to have an even number, otherwise it would be contradiction. Note that the second coordinate is an even number so that we must have 0n, and hence 00,1,2bdd. In addition, both 1b and 2b must be odd numbers. Now, 112 20 0,, ,bdb dbd implies 02dd should divide 12max ,dd. It follows that at least one of id must be equal to 2d. We may assume 12dd and thus  111,,2bdbd. Since 1b is an odd number and 01b, this shows 1100,,bd bd telling us 11,bdI, which is a contradiction. Remark 38. The converse of Proposition 37 is false. For example, consider 145,11. 9,11  is a prime ideal but not maximal. Corollary 39. Let 00,Ibd be an ideal of a lattice 1, such that 00,bd is directly below 2,nd which denotes the maximum element in the lattice. Let 0, 1L be a lattice where the partial ordering on L is defined by 01. Then the following are true and equivalent. 1) I is a prime ideal. 2) 1,I is a prime filter. 3) There is a homomorphism 1:,DI with 0II. Proof. Condition (1) follows by Proposition 37, and conditions (2) and (3) follow by lattice theory (see for example ). Proposition 40. Let 00,Ibd be an ideal of a lattice 1, and 2,nd denotes the maximum ele- ment in the lattice. Suppose that if  112 2,,bdbd I, then 11,bd I or 22,bd I. If 1, is a Boo- lean algebra, then I is maximal and 00,bd is directly below 2,nd. Proof. Since 1, is a Boolean algebra, I is maximal. If 00,,<2,nbd bdd, then 00,,bd bd . This shows 00,,bd bd. 7. Group Lattice Structure Let m and n be relatively prime integers with >1n. De- fine the group π,mn to be 112π,,, 1mnm nxyyxyxyx. Let V and W denote a solid torus and a twisted I-bundle over the Klein bottle K respectively. Recall that the prism manifold ,Mmn VW , where V is identified to W by a homeomorphism :VW defined by ,,smtnuvuv uv, where s and t are integers satisfying 1sn tm . The fundamental group of ,Mmn is π,mn. Theorem 41. Let H be a normal subgroup of π,mn. Then, either H is cyclic or H is isomorphic to π,bd for some relatively prime integers b and d satisfying the following conditions: b divides m, 1mb (mod 2), d = n and π,mbmn H, or 2d = n and 2π,mbmn H. Furthermore, there exists a realizable isomorphism  of π,mn such that if dn then ,mbHxy, J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 163and if 2dn then 2,mbHxy. Proof. Let H be a normal subgroup of π,mn. Let :,MMmn be the regular covering correspond- ing to H. Choose a component W of 1W and let 0:WWW. Since W is a twisted I-bundle over a Klein bottle K and 0:WW is a covering space, it follows that W is either TI where T is a torus or a twisted I-bundle over a Klein bottle. Note that each component of 1V is a solid torus. If W is TI, then there are two components of 1V whose boundaries are being identified with TI, and thus M is a lens space. In this case 1πMH is cyclic. If W is a twisted I-bundle over the Klein bottle, then there is only one component of 1V whose boundary is being identified with W, and hence M is a prism manifold. In this case ,MMbd for some relatively prime integers b and d. Furthermore there is a group ac- tion G on ,Mbd such that ,,MbdGM mn. Now 1K is a G-invariant Klein bottle. Hence by , the G-action is equivalent, via a homeomorphism h of ,Mbd , to either a standard 21r-action with 21mrb and nd, or a standard 22 1r-action with 21mrb and 2nd. These standard actions arise from the coverings of ,Mmn corresponding to the subgroups ,mbxy and 2,mbxy respecttively. Now h projects to a homeomorphism of ,Mmn real- izing . Theorem 42. Let 1111π,,mbbdx y and 2222π,,mbbdx y be subgroups of π,mn where 1i or 2. Then  12max ,112 2π,π,π,,mbbdb dbdxy where 12,bgcdbb and 12min ,ddd. The group generated by 11π,bd and 22π,bd is 12min ,π,,mbbd xy where 12.. ,blcmbb and 12max ,ddd. Proof. Let 12,bgcdbb. Note that we have bimbmbibyy for 1, 2i. This shows that 12max ,π,,mbbd xy is a subgroup of  112 2π,π,bdbd H. Since H contains π,bd, it follows that H is not cyclic. By Theorem 41, H is isomorphic to π,ln or π,2ln . Furthermore, b divides l and l divides ib, and since 12,bg.c.dbb, it follows that bl. If 1d or 2d is 2n, then since H is a subgroup of π,iibd , it follows by the above Theorem 41 that π,2Hln. Since π,bd is a subgroup of H, we must have 2dn showing π,bd H. We now suppose 12dd n, and thus dn and π,Hln. It follows that π,bd H. Let J be the group generated by 11π,bd and 22π,bd. Now 12min ,x is clearly a generator of J and π,bd. Since bmmbibbiyy , we have J contained in π,bd. To show π,bd is contained in J, we use the easily verifiable equation 1212..,..,mmgcdlcm bbmbb , and by using 12.. ,blcmbb we have 12.. ,mm mgcdbbb. Since there exist integers s and t such that 12mmmstbb b, we obtain 12stmm mbb byyy proving the result. Let ,mn be the collection of subgroups π,,mbnbxy of π,mn where 1 or 2. Theorem 43. ,mn is a lattice of subgroups, and there exists a lattice isomorphism 1,,mn mn which sends an element π,bd in ,mn to the element ,bd in 1,mn. Proof. If 11π,bd and 22π,bd are elements in ,mn, define  112 2π,π,bdb d if 22π,bd is a subgroup of 11π,bd . For 11π,bd and 22π,bd in ,mn, define  112 21122π,π,π,π,bdb dbdb d and 112 2π,π,bdbd to be the group generated by 11π,bd and 22π,bd . By the above Theorem 42, ,mn is a lattice. Furthermore, the map which sends an element π,bd in ,mn to the element ,bd in 1,mn is a lattice isomorphism. Corollary 44. ,mn is a distributive lattice, which is a Boolean algebra if and only if the prime decomposi- tion of m is 12kjiip. Proof. This follows by Propositions 34 and 36 and Theorem 43. J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 164 For the following propositions write 0kmpm where p is an odd prime relatively prime to 0m. Proposition 45. Let π,lnpb and π,npb be subgroups of π,mn where 1l. There exists a sur- jection :π,π,llnnpb pb. Proof. Since π,,lmlpbnpbx y and π,,mpbnpbx y , define a function :π,π,llnnpb pb by lxx and lmmpbpblyy. Clearly l preserves the first relation in π,lnpb. To show that l preserves the second relation, it suffices to show that 121lpmnyx. Write 121lps, and note that  121 222 22422lss spmmm mmmmyy yyyyy, since 41my. Thus 1221lpmn mnyxyx, showing that l is a homomorphism. Since l takes generators to generators, it is also a surjection. Proposition 46. Let 212121π,π,llnnpb pb be subgroups of π,mn where 121ll. There exist sur- jections :π,π,iilli iiinnpb pb for 1, 2i and a homomorphism 1111:π,π,nnpb pb, such that the following diagram commutes where  and  are inclusions: 222π,lnpb2l22π,npb111π,lnpb11π,npb 11π,npb1l Proof. Let 212211212121π,, ,π,llmmlpbpblnnpbxyxypb   . Note that 2b divides 1b, 21 and 21ll. By Proposition 45, there exist surjections :π,π,iilli iiinnpb pb defined by iiilxx, =liiiimmpbpblyy. Define a function 1111:π,π,nnpb pb by 11xx and 11lpmmpb pbyy where 12lll. Let 10xc and 11mpbyc, and note that the relations in this group are 11101 0ccc c and 112101npbcc. Write 21lps and observe that  11 1112122 42211 111lpsspbpbpbpbpbcc ccc , since 1411pbc. Therefore 111 11122 21010 101lnnnppb pbpbcccc cc. Clearly  preserves the other relation, showing that  is a homomorphism. Since 22lpb divides 11lpb and 2 divides 1, it fol- lows that the inclusion homomorphisms  and  are defined as follows: 2211xx and 11222121llllpbmmpbpb pbyy, 2211xx and 1221bmmbpb pbyy. One can easily check that  221 2112llxxx   and 12222 212()lllmmpmpb bpbllyyy  , which verifies that our diagram commutes. J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 165Proposition 47. 0,pmn is a sublattice of 0,kpm n, and there exists a lattice surjection 00:, ,kpm npm n induced by the the family of group homomorphisms l such that  restricted to 0,pm n is the identity. Proof. It is clear that 0,pm n is a sublattice of 0,kpm n. If 0π,,kbdpmn, then lbpb for some lk. By Proposition 45, there exists a surjec- tion :π,π,llnnpb pb. Define π,π,lnnpb pb . By the commutative dia- gram in Proposition 46, it follows that  is order pre- serving. Theorem 48. Let 12ikmjiimp be the prime de- composition. Then 12,kjiipn is a sublattice of 12,ikmjiipn, and there exists a lattice surjection 11:2,2,ikkmjjiiiipn pn induced by a family of group homomorphisms such that  restricted to 12,kjiipn is the identity. Proof. Apply 11:2, 2,iir irmmmjjirrr iriik ikppn ppn  repeatedly defined in Proposition 47 for 1rk to obtain the result where  is the compositions of those r’s. 8. Some Examples In this section we present several examples which illus- trate the main theorems. Example 49. 1315,14. This example illustrates Theorem 12 that 1315,14 is isomorphic to 1315,14. Example 50. 11155,11. This is a Boolean lattice/ algebra by Proposition 36 since 1155357 11. Prime ideals are: 3,11 , 5,11  and 7,11  Their complements are lters which are: 35,11, 21, 11 and 15,11 respectively. Example 51. 2126, 20. Since 200 (mod 4), this example illustrates Theorems 15 and 16 that  221126,20126,2063,20 . Example 52. 25040,20. This example again illus- trates Theorem 16 and also that 25040,20 is iso- morphic to 2126, 20. J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 166 Example 53. 25040,5. This illustrates Theorem 16 that 25040,5 is isomorphic to a disjoint union of isomorphic lattices  11 1163,5126,5252,5504,5 .  Example 54. 25040,10. This example illustrates Theorem 16 that 31 11263,5 63,10jj. Example 55. 35030,10. This example illustrates Theorems 18 and 19 that 3315030,105030,1063,2510 . Example 56. 4126,5. This example illustrates Theorem 21 that 41126,10 126,5. Example 57. 5630,5. This example illustrates Theorem 22 that 511630,563,5126,5. J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 167 Example 58. 55040,10. This example illustrates Theorem 22 that  5115040,101008,55040,10 . Example 59. 663,5. This example illustrates Theo- rems 19 and 23 that   61163,517, 2934, 29 . Example 60. 7126, 20. This illustrates Theorems 16 and 24 that 721126,20126,2063,20 . J. E. KALLIONGIS, R. OHASHI Copyright © 2012 SciRes. APM 168 Example 61. This is an example of “crush” to illustrate Theorem 42.  Apply 3  Apply 5 REFERENCES  J. Kalliongis and A. Miller, “Orientation Reversing Ac- tions on Lens Spaces and Gaussian Integers,” Journal of Pure and Applied Algebra, Vol. 212, No. 3, 2008, pp. 652-667. doi:10.1016/j.jpaa.2007.06.022  R. Stanley, “Enumerative Combinatorics Volume 1,” Wads- worth & Brooks/Cole, New York, 1986.  R. Ohashi, “The Isometry Groups on Prism Manifolds, Dissertation,” Saint Louis University, Saint Louis, 2005.  J. Kalliongis and R. Ohashi, “Finite Group Actions on Prism Manifolds Which Preserve a Heegaard Klein Bot- tle,” Kobe Journal of Math, Vol. 28, No. 1, 2011, pp. 69- 89.  B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order,” Cambridge Mathematical Textbooks, Cam- bridge University Press, Cambridge, 1990.