aa r X i v : . [ m a t h . R A ] M a y Arithmetic of idempotents in Z /m Z Kelly Isham and Laura Monroe ∗ Abstract
Idempotent elements are a well-studied part of ring theory, with several identitiesof the idempotents in Z /m Z already known. Although the idempotents are not closedunder addition, there are still interesting additive identities that can be derived andused.In this paper, we give several new identities on idempotents in Z /m Z . We relatefinite sublattices over Z /k Z for all integers k to an infinite lattice that is embeddedin the divisibility lattice on N and to each other as sublattices of this infinite lattice.Using this relation, we generalize several identities on idempotents in Z /m Z to thoseinvolving idempotents related to these finite sublattices.Finally, as an application of the above idempotent identities, we derive an algorithmfor calculating modular exponentiation over Z /m Z . An element d in a ring R is idempotent if d = d . In 1882, Peirce published [6], which demon-strated the important role that idempotent and nilpotent elements play in ring theory. Sincethen, researchers have tried to understand properties of idempotents in both commutativeand noncommutative rings. Many important results involve sets of idempotents that are primitive and orthogonal . An idempotent d is primitive if dR is indecomposable. A pair { d, e } of idempotents are orthogonal if de = 0 in R . In this paper, we focus on idempo-tents in R = Z /m Z . We give more attention to the idempotents that are not orthogonal orprimitive.While additive identities involving the primitive orthogonal idempotents modulo m areknown (see for example the summary paper on idempotents [7]), we have not found manyidentities involving arbitrary idempotents in the literature. In this paper, we provide severalnew identities. Further, we establish a connection between the infinite divisibility latticeand idempotents by defining consistent sublattices. We generalize the idempotent identitiesmodulo m to identities involving the elements of the consistent sublattices modulo g where g is the supremum of the given sublattice. ∗ This publication is unclassified and has been assigned LANL identifier LA-UR-20-23432. A portion of thiswork was performed at the Ultrascale Systems Research Center (USRC) at Los Alamos National Laboratory,supported by the U.S. Department of Energy contract DE-FC02-06ER25750. The first author was supportedin part by an appointment with the National Science Foundation (NSF) Mathematical Sciences GraduateInternship (MSGI) Program sponsored by the NSF Division of Mathematical Sciences. m . Severalauthors including Hewitt and Zuckerman in [3] and Schwarz in [8] have studied the orbits a, a , . . . in Z /m Z and have shown that eventually a k = d for some idempotent element d ∈ Z /m Z . In [4], we define the sequential power graph with vertex set V = Z /m Z andedges ( a, b ) such that a = c i and b = c i +1 for some c ∈ Z /m Z and i ∈ N . Also in [4], we showthat much of the structure of this graph follows directly from the idempotents. In Section6, we use some additive identities on idempotents and results about the components of thesequential power graph to provide an algorithm based on the Chinese Remainder Theoremfor computing b e (mod m ), assuming the factorization of m is known. We hope that theidentities in this paper will be useful in other applications as well. Throughout this paper, we assume that the factorization of m is known. Theorem 3.1 pro-vides the structure of idempotents modulo m . Let I = { i , . . . , i s } . To introduce notationearly on, this theorem shows that d I = a I p e i i · · · p e is i s such that d I ≡ p e j j ) for all j ∈ R \ I . The idempotent d I is determined by the set I .We define certain notations that we will use throughout this paper. • m = p e · · · p e r r is the modulus. • R = { , , . . . , r } is the set of all indices referring to the prime powers dividing m . • I = { i , . . . , i s } ⊂ R defines the primes that divide the idempotent d I . (cid:5) g I = gcd( d I , m ) = p e i i · · · p e is i s is the greatest common divisor of d I and m . (cid:5) d I = a I g I is the idempotent corresponding to I , with a I relatively prime to mg I = g R \ I . • R \ I = { i s +1 , . . . , i r } ⊂ R defines the set dual to I in the following sense (cid:5) g R \ I = gcd( d R \ I , m ) = p e is +1 i s +1 · · · p e ir i r is the greatest common divisor of d R \ I and m . (cid:5) d R \ I = a R \ I g R \ I is the idempotent corresponding to R \ I , with a R \ I relativelyprime to mg R \ I = g I . Z /m Z The idempotents of Z /m Z have been well-studied. We begin with the structure theorem foridempotents modulo m . 2 heorem 3.1. [3, Theorem 2.2] Let I = { i , . . . , i s } ⊆ R . Then the idempotents of Z /m Z are of the form a I g I (mod m ) for any a I satisfying a I g I ≡ mg I ) such that gcd ( a I , mg I ) =1 . Corollary 3.2. [3, Theorem 2.2] There are r idempotent elements in Z /m Z . It is well-known (see e.g. [7]) that d R \{ i } for i = 1 , . . . , r are the primitive orthogonalidempotents of Z /m Z . The following proposition follows immediately from this fact. Proposition 3.3. [6, Proposition 41], [7, Page 1237] The direct sum d R \{ } ( Z /m Z ) × × · · · × d R \{ r } ( Z /m Z ) × is the Peirce decomposition of the ring Z /m Z , up to order of idempotents. Further, the idempotents under multiplication are well-understood.
Proposition 3.4. [3, Corollary 2.21] The idempotents of Z /m Z form a monoid under mul-tiplication. Proposition 3.5. [3, Corollary 2.21] Let { I } be a set of subsets of R . Then Q I d I ≡ d S I (mod m ) . Remark 3.6.
All idempotents besides 1 are zero divisors, so cannot be units. Proposition3.5 shows in a different way that the all idempotents not equal to 1 cannot be units.
There is a standard partial ordering on idempotents d, e in a ring R given by d ≤ e if andonly if de = ed = e in R . If R is commutative, the idempotents form a lattice. While thereis a standard definition of the meet and join of this lattice, we will instead give an equivalentdefinition when R = Z /m Z . By Proposition 3.5, if d I and d J are idempotents in Z /m Z ,then d I d J ≡ d J (mod m ) if and only if I ∪ J = J . Therefore, we can express this partialordering in a simpler way. This partial order induces a lattice structure on the idempotentsmodulo m . Definition 3.7.
Let d I , d J be idempotents in Z /m Z . Then d I ≤ d J if and only if I ⊆ J .The idempotents in Z /m Z form a lattice with d I ∨ d J = d I ∪ J and d I ∧ d J = d I ∩ J . Theinfimum is d {} = 1 and the supremum is d R = 0. Example 3.8.
Let m = p e p e p e . Then the lattice for idempotents in Z /m Z is given by0 d { , } d { , } d { , } d { } d { } d { } { , , } under thepartial ordering ⊆ . Definition 3.9.
The k th level of the idempotent lattice is the row in the lattice consistingof elements d I where | I | = k . We say the 0 th level of the lattice is { } . An idempotent is a top-level idempotent if it is an element of the ( r − st row. Remark 3.10.
There is another way of defining the lattice of idempotents. Recall that d I ≡ g I a I (mod p ) where g I is the product of the prime powers from index set I dividing m . Define a partial ordering d I ≤ d J if and only if g I | g J . The meet is d I ∨ d J = lcm( g I , g J )and the join is d I ∧ d J = gcd( g I , g J ). This lattice is isomorphic to the subset lattice andthus to our above definition of the idempotent lattice modulo m . To see this, notice thatlcm( g I , g J ) = g I ∪ J and gcd( g I , g J ) = g I ∩ J .We introduce the lattice of idempotents here to provide motivation for our additiveidentities below. In particular, the example of an idempotent lattice will be useful as weprovide additive identities across a level of the lattice (Proposition 3.21) or across an entiresublattice (Proposition 3.24) Although the idempotents form a multiplicative monoid, and not a group or a ring, thereis still a good amount of structure to this set. Addition on idempotents is not as simple asmultiplication. For one thing, the idempotents are not closed under addition; still, there aresome additive identities that can be proved.Much is known, classically and otherwise, about the primitive orthogonal idempotentelements ( d R \{ i } ) including several of the additive identities that we cite below. However, toour knowledge, there are very few identities involving the other levels of idempotents in theliterature. We provide several new additive identities involving these other level idempotents. Proposition 3.11. [6] Let d I be the idempotent in Z /m Z defined by set I . Then − d I (mod m ) is the idempotent d R \ I . Corollary 3.12. [6] Let d I and d R \ I be idempotents in Z /m Z defined by sets I and R \ I respectively. Then d I + d R \ I ≡ m ) . Now that we have established a few basic addition formulas, we note that we can imposesubtraction on idempotents as well. Observe that division of idempotents is undefined sinceall idempotents not equal to 1 are zero divisors.
Corollary 3.13. d J − d I ≡ d J + d R \ I − m ) . Proposition 3.14. [7, Lemma 11] Let R = ∅ . Let J = { j , . . . , j s } ⊂ R and let d J beidempotent. Then P i ∈ R \ J d R \{ i } ≡ d J (mod m ) . Proposition 3.15. [8, Lemma 2.3] Let R = ∅ . Then P ri =1 d R \{ i } ≡ m ) . We now introduce an important additive identity. This will be useful in the proofs of thetheorems below. 4 roposition 3.16. d I + d J ≡ d I ∪ J + d I ∩ J (mod m ) . Proof.
By Proposition 3.14, we have d I ≡ X i ∈ R \ I d R \{ i } (mod m ) ≡ X i ∈ R \ ( I ∩ J ) d R \{ i } − X i ∈ I \ J d R \{ i } (mod m )Focusing on the last term, by properties of sets we obtain X i ∈ I \ J d R \{ i } ≡ X i ∈ R d R \{ i } − X i ∈ R \ ( I ∪ J ) d R \{ i } − X i ∈ J d R \{ i } (mod m ) ≡ − d I ∪ J − d R \ J (mod m )Putting these together, d I ≡ d I ∩ J − (cid:0) − d I ∪ J − d R \ J (cid:1) (mod m ) ≡ d I ∩ J − d I ∪ J + d R \ J (mod m )Therefore we obtain d I + d J ≡ d I ∩ J − d I ∪ J + d R \ J + d J ≡ d I ∩ J + d I ∪ J (mod m )by Proposition 3.11. Corollary 3.17.
Let d I and d J be idempotent. If m is odd, d I + d J is idempotent if andonly if d I + d J ≡ d I ∩ J (mod m ) .Proof. Observe that ( d I + d J ) ≡ d I + d J if and only if 2 d I d J ≡ m ). Since m is odd, d I + d J is idempotent if and only if d I d J ≡ m ). This occurs if and only if d I ∪ J ≡ m ). Proposition 3.18. If I , . . . , I k are pairwise disjoint and I ∪ · · · ∪ I k = J , then P ki =1 d I i ≡ k − d J (mod m ) .Proof. We will prove this by induction on k .If k = 1, then d I ≡ − d I (mod m ).Suppose P k − i =1 d I i ≡ ( k −
2) + d J ′ (mod m ) where J ′ = I ∪ · · · ∪ I k − . Then k X i =1 d I i ≡ ( k −
2) + d J ′ + d I k ≡ k − d J (mod m )by Proposition 3.16 and since J ′ ∩ I k = ∅ and J ′ ∪ I k = J .5 orollary 3.19. If I , . . . , I k are pairwise disjoint and I ∪· · ·∪ I k = R , then P ki =1 d I i ≡ k − m ) . Corollary 3.20.
Let I ⊆ R with | I | = k > . We have P i ∈ I d { i } ≡ k − d I (mod m ) . The following theorem provides additive identities for the k th level of idempotents (all d J such that | J | = k ) for all k = 0 , . . . , r − Theorem 3.21.
Let | R | = r > , and let ≤ k < r . Then P | J | = k d J ≡ (cid:0) r − k (cid:1) (mod m ) .Proof. For k = r −
1, this proposition is equivalent to Proposition 3.15.Let k < r −
1. If | J | = k , d J ≡ P d I (mod m ), where I = R \ { i } for each i ∈ R \ J , byProposition 3.14. These I are the ( r − R such that J ⊂ I , so each d I is a summand of exactly (cid:0) r − k (cid:1) of the d J . This means that the summation P d I = 1 occurs (cid:0) r − k (cid:1) times in P | J | = k d J , so P | J | = k d J = (cid:0) r − k (cid:1) . Lemma 3.22.
Suppose | I | = k . We sum over all d J so that | J | = k − n and J ⊂ I where < n < k . In other words, we sum over all elements in the idempotent lattice n levels below d I that eventually connect to d I . This gives X | J | = k − nJ ⊂ I d J ≡ (cid:18) k − n (cid:19) + (cid:18) k − n − (cid:19) d I (mod m ) . Proof. X | J | = k − nJ ⊂ I d J = X | J | = k − nJ ⊂ I X i ∈ R \ J d R \{ i } by Proposition 3.14. Note that X i ∈ R \ J d R \{ i } = X i ∈ R \ I d R \{ i } + X i ∈ ( I \ J ) d R \{ i } and that | I \ J | = n . Further, there are (cid:0) kn (cid:1) subsets J satisfying the conditions. Therefore X | J | = k − nJ ⊂ I d J ≡ (cid:18) kn (cid:19) X i ∈ R \ I d R \{ i } + X | J | = k − nJ ⊂ I X i ∈ ( I \ J ) d R \{ i } ≡ (cid:18) kn (cid:19) d I + X | J | = k − nJ ⊂ I X i ∈ ( I \ J ) d R \{ i } We now focus on understanding this last summation. When taking the summation of | J | = k − n, J ⊂ I , every term in the summation will contain all d R \{ i } such that i I . That is,6very term will contain d R \ I . Each element in R \ { i } for i ∈ I occurs ( kn ) ( k − n ) k times sincethere are (cid:0) kn (cid:1) terms, each containing k − n terms from I and all occur equally often. Thus, X | J | = k − nJ ⊂ I X i ∈ ( I \ J ) d R \{ i } = (cid:0) kn (cid:1) ( k − n ) k d R \ I Putting these together, X | J | = k − nJ ⊂ I d J ≡ (cid:18) kn (cid:19) d I + (cid:0) kn (cid:1) ( k − n ) k d R \ I ≡ (cid:18) kn (cid:19) − (cid:0) kn (cid:1) ( k − n ) k ! d I + (cid:0) kn (cid:1) ( k − n ) k ≡ (cid:18) k − n − (cid:19) d I + (cid:18) k − n (cid:19) (mod m ) . The last lemma seems abstract, so consider the following corollary when n = 1. Corollary 3.23.
Suppose | I | = k . We sum over all d J so that | J | = k − and J ⊂ I . Inother words, we sum over all elements in the idempotent lattice in the level directly below d I that are connected to d I . This gives the identity X | J | = k − ,J ⊂ I d J ≡ ( k −
1) + d I (mod m ) . With Lemma 3.22 established, we can now determine the summation of all idempotentsbelow d I in terms of I only. We state the theorem modulo m and provide a corollary fordetermining the summation modulo g I as well. Theorem 3.24.
Let | I | = k . Then P J ⊆ I d J ≡ k − (1 + d I ) (mod m ) .Proof. By Lemma 3.22, X | J | = k − nJ ⊂ I d J ≡ (cid:18) k − n (cid:19) + (cid:18) k − n − (cid:19) d I (mod m ) . When J = ∅ , the idempotent is d J = 1 and when J = I , the idempotent is d J = d I .7herefore X J ⊆ I d J ≡ d I + k − X n =1 X | J | = k − nJ ⊂ I d J (mod m ) ≡ d I + k − X n =1 (cid:18) k − n (cid:19) + d I k − X n =1 (cid:18) k − n − (cid:19) (mod m ) ≡ k − + d I k − X n =0 (cid:18) k − n (cid:19) (mod m ) ≡ k − + 2 k − d I (mod m ) Corollary 3.25.
Let | I | = k . Then P J ⊆ I d J ≡ k − (mod g I ) . Corollary 3.26. P I ⊆ R d I ≡ r − (mod m ) . In Section 3.2, we defined the lattice of idempotents in Z /m Z . In this section, we seekto generalize the notion of an idempotent lattice. By Remark 3.10, the idempotent latticemodulo m can be defined in terms of the elements g I = gcd( d I , m ) for each I ⊆ R . Wegeneralize this definition of the idempotent lattice modulo m to both finite and infinitelattices. These will be subsets of the infinite divisibility lattice on N , which we call consistentsublattices . Since the only idempotents in Z are 0 and 1, the notion of an infinite idempotentlattice in N does not make sense. However, the construction of the infinite divisibility latticeon N permits us to relate an infinite set of finite lattices over Z /m Z for integers m to eachother as sublattices of this infinite lattice. m Definition 4.1.
The infinite divisibility lattice on N is given by the partial ordering a ≤ b if and only if a | b . The meet of two elements a, b is a ∨ b = lcm( a, b ) and the join is a ∧ b =gcd( a, b ). Definition 4.2.
List the primes in increasing order with p = 2. Let S ⊆ N and T ⊆ S .Let E be an infinite sequence E = ( e , e , . . . ) of elements in N ∪ { } so that e i > i ∈ S and e i = 0 for all i S . A sublattice of the infinite divisibility lattice is consistent ifthe elements of this lattice are exactly the integers g K = Q k ∈ K p e k k for all T ⊆ K ⊆ S . Wedenote this sublattice L E,S,T . Remark 4.3.
Suppose S = { s , . . . , s r } is finite and T = ∅ . Notice that E contains onlyfinitely many nonzero elements. Set m = p e s s · · · p e sr s r and let d K be the idempotent modulo m corresponding to set K ⊆ S . The consistent sublattice defined by sets E , S and T = ∅ g K = gcd( m, d K ). Therefore L E,S,T is isomorphic to the asublattice of the idempotent lattice modulo m . Since g K depends only on the sets E, S , and T , it makes sense to extend the notion of the elements g K when S is infinite.Suppose S is infinite, T ⊆ S , and E = ( e , e , . . . ) with e i > i ∈ S and e i = 0 if i S . Write the supremum g S = Q s ∈ S p e s s and the infimum g T = Q t ∈ T p e t t . Let L E,S,T be the consistent sublattice with supremum g S and infimum g T . Then b ∈ L if and only if b = g T Q j ∈ K p e j j for some K ⊆ S \ T. Definition 4.4.
A consistent sublattice of the infinite divisibility lattice is maximally con-sistent if S = N and T = ∅ , that is, if the supremum is divisible by every prime and theinfimum is 1.We can uniquely determine a maximally consistent sublattice from the first level, whichconsists of prime powers. It is clear that there is a bijection between sequences of positiveintegers and maximally consistent sublattices; each sequence of positive integers ( e , e , . . . )determines a distinct maximally consistent sublattice by setting the first level equal to { p e , p e , . . . } and this correspondence is surjective. Proposition 4.5.
The set of maximally consistent sublattices of L is uncountable and hascardinality of R .Proof. The set of maximally consistent sublattices of L has the cardinality of R . From theremark above, we know that each maximally consistent sublattice is uniquely determinedby the sequence ( e , e , . . . ) of positive integers. There is a bijection of the set of suchsequences to R by sending ( e , e , . . . ) to . ( e − e − · · · . The inverse of this function is .b b . . . ( b + 1 , b + 1 , . . . ). Remark 4.6.
Let the idempotent lattice modulo m be denoted L m . Observe that there isa bijection between finite maximally consistent lattices and lattices of the form L m . Let L E,S,T be a finite consistent lattice and let S ′ ⊇ S be a finite set. Then L E,S,T can beextended to the finite maximally consistent lattice L E,S ′ , ∅ . As noted in the remark above, L E,S ′ , ∅ is isomorphic to the lattice L m of idempotents in Z /m Z where m = g S ′ is the supre-mum of L E,S ′ ,T . Therefore finite consistent sublattices are isomorphic to sublattices of L m .We denote this sublattice of L m by L m,S,T since the sequence E is clear given m .In the next section, we will consider idempotent identities on sublattices of L m . We usenotions from consistent lattices throughout. Many of the theorems we have proved in Section 3 have analogues in the consistent lattices L m,S,T of L m . For notation, we set g S to be the supremum of L m,S,T and we let d S bethe corresponding idempotent modulo m . Recall that m = g S ′ for some finite set S ′ ⊇ S .Similarly, g T is the infimum and d T is the corresponding idempotent modulo m .9 .1 Multiplication on idempotents in finite consistent lattices Proposition 5.1.
Let L m,S,T be a sublattice of L m . Let { I } be a set of subsets of S thatcontain T . Then Q I d I ≡ d S I (mod g S ) . Proof.
Since g S | m and this equivalence holds (mod m ), then this equivalence must also hold(mod g S ). In this section, we generalize the additive identities established in Section 3.3. Observe thatfor any element g I ∈ L m,S,T , T ⊆ I ⊆ S . Therefore all sets considered in the followingpropositions and theorems must contain T . Proposition 5.2.
Let L m,S,T be a sublattice of L m and let T ⊆ I, J ⊆ S . Then d I + d J ≡ d I ∪ J + d I ∩ J (mod g S ) . Proof.
This holds modulo m and g S | m . Proposition 5.3.
Let L m,S,T ⊆ L m . Let I , . . . , I k be sets such that T ⊆ I ℓ ⊆ S for all ≤ ℓ ≤ k and I ℓ ∩ I j = T for all ℓ = j . Suppose that I ∪ · · · ∪ I k = J . We have P ki =1 d I i ≡ ( k − d T + d J (mod g S ) .Proof. We will prove this by induction on k .If k = 1, then d I ≡ (1 − d T + d I (mod g S ).Let J ′ = I ∪ · · · ∪ I k − and assume that P k − i =1 d I i ≡ ( k − d T + d J ′ (mod g S ). Observethat J ′ ∩ I k = T and J ′ ∪ I k = J . Applying Proposition 5.2 gives k X i =1 d I i ≡ ( k − d T + d J ′ + d I k ≡ ( k − d T + d T + d J (mod g S ) . Corollary 5.4.
Let L m,S,T be a sublattice of L m . If I , . . . , I k are sets such that T ⊆ I ℓ ⊆ S for all ≤ ℓ ≤ k , I ℓ ∩ I j = T for all ℓ = j , and I ∪ · · · ∪ I k = S , then P ki =1 d I i ≡ ( k − d T (mod g S ) . Corollary 5.5.
Let L m,S,T be as above. Let T ⊆ I ⊆ S with | I | = k > . Then P i ∈ I d T ∪{ i } ≡ ( k − d T + d I (mod g S ) . The following corollary is a generalization of Corollary 3.12. Here, the infimum is d T ,so if g I = gcd( d I , g S ) and g J = gcd( d J , g S ) are elements in L m,S,T , we require T ⊆ J aswell. Therefore, J = S \ I does not make sense in this setting. The correct generalization is J = ( S \ I ) ∪ T = S \ ( I \ T ) . Corollary 5.6.
Let T ⊆ I ⊆ S and L m,S,T ⊆ L m . We have d I + d S \ ( I \ T ) ≡ d T (mod g S ) . emma 5.7. Suppose S = ∅ . Let L m,S,T ⊆ L m , and T ⊂ I ⊆ S . Then P i ∈ I \ T d S \{ i } ≡ d S \ ( I \ T ) (mod g S ) .Proof. We will prove the lemma using induction on | I | > | T | . If I = T ∪ { i } for some i ∈ S \ T , then X i ∈ I \ T d S \{ i } = d S \{ i } = d S \ ( I \ T ) . Now suppose that this equivalence holds for | I ′ \ T | = k −
1. Then P i ∈ I ′ \ T d S \{ i } ≡ d S \ ( I ′ \ T ) (mod g S ). Pick an element i k ∈ S \ T so that i k I ′ and set I = I ′ ∪ { i k } . Then X i ∈ I \ T d S \{ i } = X i ∈ I ′ \ T d S \{ i } + d S \{ i k } ≡ d S \ ( I ′ \ T ) + d S \{ i k } (mod g S ) ≡ d S + d S \ ( I \ T ) (mod g S ) ≡ d S \ ( I \ T ) (mod g S ) . The penultimate line is due to Proposition 5.2. The last line holds because g S | d S . Proposition 5.8.
Suppose S = ∅ and let L m,S,T be as above. If T ⊆ J ⊂ S , P i ∈ S \ J d S \{ i } ≡ d J (mod g S ) .Proof. Use Lemma 5.7 and set I = S \ J . Proposition 5.9.
Let L m,S,T be as above with | S | = s ≥ and | T | = t . Let t ≤ k < s . Then P | J | = k d J ≡ (cid:0) s − t − k − t (cid:1) d T (mod g S ) .Proof. Recall that T ⊆ J ⊂ S for all J so that g J ∈ L m,S,T . If | J | = k , d J ≡ P i ∈ S \ J d S \{ i } (mod g S ), by Proposition 5.8. The terms S \ { i } are the ( s − S with i ∈ S \ T . Notice that each d S \{ i } is in exactly (cid:0) s − t − k − t (cid:1) of the terms in the summation. ByProposition 5.8, X i ∈ S \ T d S \{ i } ≡ d T (mod g S ) . Therefore the summation reduces to X | J | = k d J ≡ (cid:18) s − t − k − t (cid:19) d T (mod g S ) . Lemma 5.10.
Let S = ∅ , L m,S,T ⊆ L m . Suppose T ⊂ I ⊂ S and set | T | = t, | I | = k . Wesum over all d J so that | J | = k − n and T ⊂ J ⊂ I where < n < k − t . In other words,we sum over all elements in the idempotent lattice n levels below d I that eventually connectto d I . This gives the identity X | J | = k − nT ⊂ J ⊂ I d J ≡ (cid:18) k − t − n (cid:19) d T + (cid:18) k − t − n − (cid:19) d I (mod g S ) . roof. By Proposition 5.8, X | J | = k − nT ⊂ J ⊂ I d J = X | J | = k − nT ⊂ J ⊂ I X i ∈ S \ J d S \{ i } . The inner sum splits up further into X i ∈ S \ J d S \{ i } = X i ∈ S \ I d S \{ i } + X i ∈ ( I \ J ) d S \{ i } and that | I \ J | = n . Further, there are (cid:0) k − tn (cid:1) subsets J satisfying the conditions T ⊂ J ⊂ I such that | I \ J | = n . Therefore X | J | = k − nT ⊂ J ⊂ I d J ≡ (cid:18) k − tn (cid:19) X i ∈ S \ I d S \{ i } + X | J | = k − nT ⊆ J ⊂ I X i ∈ ( I \ J ) d S \{ i } ≡ (cid:18) k − tn (cid:19) d I + X | J | = k − nT ⊂ J ⊂ I X i ∈ ( I \ J ) d S \{ i } Consider this last summation. The terms are the d S \{ i } such that i ∈ I \ T . Each element d s for s ∈ S \ { i } with i ∈ I occurs ( k − tn ) ( k − t − n ) k − t times. This is true as there are (cid:0) k − tn (cid:1) subsets J and each such subset contains k − t − n terms from I \ T . All terms occur equally oftenand we have overcounted by a factor of k − t . Notice that X i ∈ I \ T d S \{ i } ≡ d S \ ( I \ T ) by Lemma 5.7. Thus X | J | = k − nT ⊂ J ⊂ I X i ∈ ( I \ J ) d S \{ i } = (cid:0) k − tn (cid:1) ( k − t − n ) k − t d S \ ( I \ T ) . Putting these summations together, X | J | = k − nT ⊂ J ⊂ I d J ≡ (cid:18) k − tn (cid:19) d I + (cid:0) k − tn (cid:1) ( k − t − n ) k − t d S \ ( I \ T ) ≡ (cid:18) k − tn (cid:19) − (cid:0) k − tn (cid:1) ( k − t − n ) k − t ! d I + (cid:0) k − tn (cid:1) ( k − t − n ) k − t d T ≡ (cid:18) k − t − n − (cid:19) d I + (cid:18) k − t − n (cid:19) d T (mod g S ) . n = 1. Corollary 5.11.
Let L m,S,T be a sublattice as above and let | I | = k . We sum over all d J sothat | J | = k − and T ⊂ J ⊂ I . In other words, we sum over all elements in the idempotentlattice in the level directly below d I that are connected to d I . We find X | J | = k − ,T ⊂ J ⊂ I d J ≡ ( k − t − d T + d I (mod g S ) . The following theorem gives a formula for the summation of all idempotents below d I that is in terms of I, S , and T only. Theorem 5.12.
Let L m,S,T ⊆ L m . Let | I | = k . Then P T ⊆ J ⊆ I d J ≡ k − t − ( d T + d I )(mod g S ) .Proof. By Lemma 5.10, X | J | = k − nT ⊂ J ⊂ I d J ≡ (cid:18) k − t − n (cid:19) d T + (cid:18) k − t − n − (cid:19) d I (mod g S ) . When J = T , we see that d J = d T and when J = I , d J = d I . Therefore X T ⊆ J ⊆ I d J ≡ d T + d I + k − t − X n =1 X | J | = k − nT ⊂ J ⊂ I d J (mod g S ) ≡ d T + d I + d T k − t − X n =1 (cid:18) k − t − n (cid:19) + d I k − t − X n =1 (cid:18) k − t − n − (cid:19) (mod g S ) . The result follows from identities on binomial coefficients. X T ⊆ J ⊆ I d J ≡ k − t − d T + 2 k − t − d I (mod g S ) . Corollary 5.13.
Let L m,S,T ⊆ L m . Let | I | = k . We find P T ⊆ J ⊆ I d J ≡ k − t − d T (mod g I ) . Corollary 5.14. If L m,S,T ⊆ L m , P T ⊆ J ⊆ S d J ≡ k − t − d T (mod g S ) . Z /m Z Consider the sequence a, a , a , . . . in Z /m Z . The sequence must be finite, so eventually a k = a ℓ . We call a, a , . . . , a k − the tail and a k , . . . , a ℓ the cycle corresponding to a .13 efinition 6.1. The sequential power graph is the graph with vertex set V = Z /m Z anddirected edges ( a, b ) if and only if a ≡ c i (mod m ) and b ≡ c i +1 (mod m ) for some c ∈ Z /m Z and some i ∈ N .This graph represents the connection between these sequences a, a , a , . . . for each a ∈ Z /m Z . It is closely related to the power graph, which was defined in [5]. The power graphhas been well-studied, see for example [1] and [2]. Much of the structure of this graph followsdirectly from the structure of the idempotents modulo m . We summarize the connection toidempotents in the below propositions from [4]. This connection will be used in Section 6.2. Proposition 6.2. [4, Corollary 5.17, Theorem 5.18] The connected components of the se-quential power graph are composed of the roots of the idempotents in Z /m Z . There is a one-to-one correspondence between connected components and idempotents of Z /m Z . Thereforewe can refer to a component C I as the component with idempotent d I . Proposition 6.3. [4, Theorem 5.21] The cycles of connected components are themselves aconnected subgraph, and are groups under multiplication. The associated idempotent d is thegroup’s unity. Further, the elements in the cycles of component C I are exactly the elementsin d I U where U = ( Z /m Z ) × . Let a, a , . . . , a k = d be the sequence of distinct powers of a ∈ Z /m Z . Observe that if a is a cycle element, then a k +1 = a . Whenever a is in a cycle, the above propositions implythat a ∈ d I U , that is, a = d I u for some u ∈ U . The tail elements cannot be determinedusing d I , but can be expressed in terms of a similar element. Definition 6.4. [4] Let I = { i , . . . , i s } ⊆ R . The multiplier corresponding to set I is π I = p i · · · p i s . Proposition 6.5. [4, Proposition 5.2] Let π I be the multiplier corresponding to set I ⊆ R .The elements of C I are the integers π I x , where gcd ( x, mg I ) = 1 . First, suppose that u ∈ U = ( Z /m Z ) × and we wish to compute u e (mod m ). We give analgorithm for computing u e using the structure of the idempotents from Section 3 which isa reformulation of the Chinese Remainder Theorem. In particular, the following theoremshows a connection between primitive orthogonal idempotents modulo m and the ChineseRemainder Theorem. This connection is known, but not widely; see [7] for a history on therelation between idempotents and the Chinese Remainder Theorem. Further, this theoremis true in general for any b ∈ Z /m Z , however we state it just for the units in Z /m Z to makea distinction between this and the next theorem. Theorem 6.6. [7] Let u ∈ U , the set of units, and let d i = d R \{ i } be the top-level idempotents.Let e ∈ N . Then u e ≡ r X i =1 d i u e (mod φ ( p eii )) (mod m ) .
14e now generalize the algorithm to compute b e (mod m ) for any b ∈ C I where | I | < r − b e (mod m ) for C I where | I | = r −
1. That is, when the idempotent d I is a top-level idempotent. We introduce twotheorems that are reductions of the previous theorem. We provide proofs of these theoremsthat rely on an identity from Section 5.2 and results from Section 6.1. To the best of ourknowledge, these reductions are not used in practice. Theorem 6.7.
Let b ∈ C I so that b = b k for some k > . In other words, d I b = b . Let T = I and S = R . Let d i = d S \{ i } . Then b e ≡ X i ∈ S \ T d i b e (mod φ ( p eii )) (mod m ) . Proof.
Since S = R , then g S = m . By Proposition 5.8, d T ≡ X i ∈ S \ T d i (mod m ) . Therefore, b e ≡ d T b e ≡ X i ∈ S \ T d i b e (mod m ) . Since each d i contains all primes in the factorization of mp eii , then d i ≡ p e j j ) for all j = i and d i ≡ p e i i ) by Theorem 3.1. Thus d i b e ≡ d i b e (mod φ ( p eii )) (mod m ) for each i ∈ S \ T .The previous theorem only holds when b is in the cycle of C I . The benefit to this previoustheorem is it has no restrictions on the exponent e . In the following, we show that the sameresult holds when b is not in the cycle, but only when e is large enough. Theorem 6.8.
Set S = R . Let T ⊆ S . Let b ∈ C T , that is, let b = π T x for some x relatively prime to mg T . Let d j = d S \{ j } be the top-level idempotents, for ≤ j ≤ r . Let e ≥ max( e , . . . , e r ) . Then b e ≡ X i ∈ S \ T d i b e (mod φ ( p eii )) (mod m ) . Proof.
Since e ≥ max( e , . . . , e r ) and b e = π eT x e , then d T | π e . Thus d T | b e and so d T b e ≡ b e (mod m ). The result follows by applying Theorem 6.7. Remark 6.9.
Note that we may have e (mod φ ( p e i i )) = 0. In this case, b is considered asan element in d i U . Since the identity of d i U is d i , then here b = d i . Remark 6.10.
We have throughout used the Euler totient function. We can, however, usethe Carmichael totient function for a tighter calculation.15
Acknowledgements
A portion of this work was performed at the Ultrascale Systems Research Center (USRC)at Los Alamos National Laboratory, supported by the U.S. Department of Energy contractDE-FC02-06ER25750. The first author was supported in part by an appointment with theNational Science Foundation (NSF) Mathematical Sciences Graduate Internship (MSGI)Program sponsored by the NSF Division of Mathematical Sciences.
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