w.. vsnth kndsmy smrndche semirings, semifields, nd semivector spces {,,c,d} {,,c} {,,d} {,d,c} {,d,c} {,} {,c} {,d} {,c} {,d} {d,c} {} {} {c} {d} {φ} mericn reserch press rehooth 22
Smrndche Semirings, Semifields, nd Semivector spces W. B. Vsnth Kndsmy Deprtment of Mthemtics Indin Institute of Technology, Mdrs Chenni 636, Indi Americn Reserch Press Rehooth, NM 22
The picture on the cover is Boolen lger constructed using the power set P(X) where X = {,, c} which is finite Smrndche semiring of order 6. This ook cn e ordered in pper ound reprint from: Books on Demnd ProQuest Informtion & Lerning (University of Microfilm Interntionl) 3 N. Zee Rod P.O. Box 346, Ann Aror MI 486-346, USA Tel.: -8-52-6 (Customer Service) http://wwwli.umi.com/od/ nd online from: Pulishing Online, Co. (Settle, Wshington Stte) t: http://pulishingonline.com This ook hs een peer reviewed nd recommended for puliction y: Prof. Murtz A. Qudri, Dept. of Mthemtics, Aligrh Muslim University, Indi. Prof. B. S. Kirngi, Dept. of Mthemtics, Mysore University, Krntk, Indi. Prof. R. C. Agrwl, Dept. of Mthemtics, Lucknow University, Indi. Copyright 22 y Americn Reserch Press nd W. B. Vsnth Kndsmy Rehooth, Box 4 NM 87322, USA Mny ooks cn e downloded from: http://www.gllup.unm.edu/~smrndche/ebooks-otherformts.htm ISBN: -93233-62-4 Stndrd Address Numer: 297-592 Printed in the United Sttes of Americ 2
Printed in the United Sttes of Americ CONTENTS Prefce 5. Preliminry notions. Semigroups, groups nd Smrndche semigroups 7.2 Lttices.3 Rings nd fields 9.4 Vector spces 22.5 Group rings nd semigroup rings 25 2. Semirings nd its properties 2. Definition nd exmples of semirings 29 2.2 Semirings nd its properties 33 2.3 Semirings using distriutive lttices 38 2.4 Polynomil semirings 44 2.5 Group semirings nd semigroup semirings 46 2.6 Some specil semirings 5 3. Semifields nd semivector spces 3. Semifields 53 3.2 Semivector spces nd exmples 57 3.3 Properties out semivector spces 58 4. Smrndche semirings 4. Definition of S-semirings nd exmples 65 4.2 Sustructures in S-semirings 67 4.3 Smrndche specil elements in S-semirings 75 4.4 Specil S-semirings 8 4.5 S-semirings of second level 86 4.6 Smrndche-nti semirings 9 5. Smrndche semifields 5. Definition nd exmples of Smrndche semifields 95 5.2 S-wek semifields 97 5.3 Specil types of S-semifields 98 5.4 Smrndche semifields of level II 99 5.5 Smrndche nti semifields 3
6. Smrndche semivector spces nd its properties 6. Definition of Smrndche semivector spces with exmples 3 6.2 S-susemivector spces 4 6.3 Smrndche liner trnsformtion 6 6.4 S-nti semivector spces 9 7. Reserch Prolems Index 3 4
PREFACE Smrndche notions, which cn e undoutedly chrcterized s interesting mthemtics, hs the cpcity of eing utilized to nlyse, study nd introduce, nturlly, the concepts of severl structures y mens of extension or identifiction s sustructure. Severl reserchers round the world working on Smrndche notions hve systemticlly crried out this study. This is the first ook on the Smrndche lgeric structures tht hve two inry opertions. Semirings re lgeric structures with two inry opertions enjoying severl properties nd it is the most generlized structure for ll rings nd fields re semirings. The study of this concept is very megre except for very few reserch ppers. Now, when we study the Smrndche semirings (S-semiring), we mke the richer structure of semifield to e contined in n S-semiring; nd this S-semiring is of the first level. To hve the second level of S-semirings, we need still richer structure, viz. field to e suset in S-semiring. This is chieved y defining new notion clled the Smrndche mixed direct product. Likewise we lso define the Smrndche semifields of level II. This study mkes one relte, compre nd contrsts weker nd stronger structures of the sme set. The motivtion for writing this ook is two-fold. First, it hs een our im to give n insight into the Smrndche semirings, semifields nd semivector spces. Secondly, in order to mke n orgnized study possile, we hve lso included ll the concepts out semirings, semifields nd semivector spces; since, to the est of our knowledge we do not hve ooks, which solely dels with these concepts. This ook introduces severl new concepts out Smrndche semirings, semifields nd semivector spces sed on some pper y F. Smrndche out lgeric structures nd nti-structures. We ssume t the outset tht the reder hs strong ckground in lger tht will enle one to follow nd understnd the ook completely. This ook consists of seven chpters. The first chpter introduces the sic concepts which re very essentil to mke the ook self-contined. The second chpter is solely devoted to the introduction of semirings nd its properties. The notions out semifields nd semivector spces re introduced in the third chpter. Chpter four, which is one of the mjor prts of this ook contins complete systemtic introduction of ll concepts together with sequentil nlysis of these concepts. Exmples re provided undntly to mke the strct definitions nd results esy nd explicit to the reder. Further, we hve lso given severl prolems s exercises to the student/ resercher, since it is felt tht tckling these reserch prolems is one of the wys to get deeply involved in the study of Smrndche semirings, semifields nd semivector spces. The fifth chpter studies Smrndche semifields nd elortes some of its properties. The concept of Smrndche semivector spces re treted nd nlysed in 5
chpter six. The finl chpter includes 25 reserch prolems nd they will certinly e oon to ny resercher. It is lso noteworthy to mention tht t the end of every chpter we hve provided iliogrphicl list for supplementry reding, since referring to nd knowing these concepts will equip nd enrich the resercher's knowledge. The ook lso contins comprehensive index. Finlly, following the suggestions nd motivtions of Dr. Florentin Smrndche s pper on Anti-Structures we hve introduced the Smrndche nti semiring, nti semifield nd nti semivector spce. On his suggestion I hve t ech stge introduced II level of Smrndche semirings nd semifields. Overll in this ook we hve totlly defined 65 concepts relted to the Smrndche notions in semirings nd its generliztions. I deeply cknowledge my children Meen nd Km whose joyful persusion nd support encourged me to write this ook. References:. J. Cstillo, The Smrndche Semigroup, Interntionl Conference on Comintoril Methods in Mthemtics, II Meeting of the project 'Alger, Geometri e Comintori', Fculdde de Ciencis d Universidde do Porto, Portugl, 9- July 998. 2. R. Pdill, Smrndche Algeric Structures, Smrndche Notions Journl, USA, Vol.9, No. -2, 36-38, (998). 3. R. Pdill. Smrndche Algeric Structures, Bulletin of Pure nd Applied Sciences, Delhi, Vol. 7 E, No., 9-2, (998); http://www.gllup.unm.edu/~smrndche/alg-s-txt.txt 4. F. Smrndche, Specil Algeric Structures, in Collected Ppers, Vol. III, Add, Orde, 78-8, (2). 5. Vsnth Kndsmy, W. B. Smrndche Semirings nd Semifields, Smrndche Notions Journl, Vol. 7, -2-3, 88-9, 2. http://www.gllup.unm.edu/~smrndche/semirings.pdf 6
CHAPTER ONE PRELIMINARY NOTIONS This chpter gives some sic notions nd concepts used in this ook to mke this ook self-contined. The serious study of semirings is very recent nd to the est of my knowledge we do not hve mny ooks on semirings or semifields or semivector spces. The purpose of this ook is two-fold, firstly to introduce the concepts of semirings, semifields nd semivector spces (which we will shortly sy s semirings nd its generliztions), which re not found in the form of text. Secondly, to define Smrndche semirings, semifields nd semivector spces nd study these newly introduced concepts. In this chpter we recll some sic properties of semigroups, groups, lttices, Smrndche semigroups, fields, vector spces, group rings nd semigroup rings. We ssume t the outset tht the reder hs good ckground in lger.. Semigroups, Groups nd Smrndche Semigroups In this section we just recll the definition of these concepts nd give rief discussion out these properties. DEFINITION..: Let S e non-empty set, S is sid to e semigroup if on S is defined inry opertion such tht. For ll, S we hve S (closure). 2. For ll,, c S we hve ( ) c = ( c) (ssocitive lw), We denote y (S, ) the semigroup. DEFINITION..2: If in semigroup (S, ), we hve = for ll, S we sy S is commuttive semigroup. If the numer of elements in the semigroup S is finite we sy S is finite semigroup or semigroup of finite order, otherwise S is of infinite order. If the semigroup S contins n element e such tht e = e = for ll S we sy S is semigroup with identity e or monoid. An element x S, S monoid is sid to e invertile or hs n inverse in S if there exist y S such tht xy = yx = e. DEFINITION..3: Let (S, ) e semigroup. A non-empty suset H of S is sid to e susemigroup of S if H itself is semigroup under the opertions of S. 7
DEFINITION..4: Let (S, ) e semigroup, non-empty suset I of S is sid to e right idel of S if I is susemigroup of S nd for ll s S nd i I we hve is I. Similrly one cn define left idel in semigroup. We sy I is n idel of semigroup if I is simultneously left nd right idel of S. DEFINITION..5: Let (S, ) nd (S, ο) e two semigroups. We sy mp φ from (S, ) (S,ο) is semigroup homomorphism if φ (s s 2 ) = φ(s ) ο φ(s 2 ) for ll s, s 2 S. Exmple..: Z 9 = {,, 2,, 8} is commuttive semigroup of order nine under multipliction modulo 9 with unit. Exmple..2: S = {, 2, 4, 6, 8, } is semigroup of finite order, under multipliction modulo 2. S hs no unit ut S is commuttive. Exmple..3: Z e the set of integers. Z under usul multipliction is semigroup with unit of infinite order. Exmple..4: 2Z = {, ±2, ±4,, ±2n } is n infinite semigroup under multipliction which is commuttive ut hs no unit. Exmple..5: Let S 2 2 =,,c,d Z4. S 2 2 is finite non-commuttive c d semigroup under mtrix multipliction modulo 4, with unit I 2 2 =. Exmple..6: Let M 2 2 =,,c,d Q, thefieldof rtionls. M 2 2 is c d non-commuttive semigroup of infinite order under mtrix multipliction with unit I 2 2 =. Exmple..7: Let Z e the semigroup under multipliction pz = {, ± p, ± 2p, } is n idel of Z, p ny positive integer. Exmple..8: Let Z 4 = {,, 2,, 3} e the semigroup under multipliction. Clerly I = {, 7} nd J = {, 2, 4, 6, 8,, 2} re idels of Z 4. Exmple..9: Let X = {, 2, 3,, n} where n is finite integer. Let S (n) denote the set of ll mps from the set X to itself. Clerly S (n) is semigroup under the composition of mppings. S(n) is non-commuttive semigroup with n n elements in it; in fct S(n) is monoid s the identity mp is the identity element under composition of mppings. Exmple..: Let S(3) e the semigroup of order 27, (which is for n = 3 descried in exmple..9.) It is left for the reder to find two sided idels of S(3). 8
Nottion: Throughout this ook S(n) will denote the semigroup of mppings of ny set X with crdinlity of X equl to n. Order of S(n) is denoted y ο (S(n)) or S(n) nd S(n) hs n n elements in it. Now we just recll the definition of group nd its properties. DEFINITION..6: A non-empty set of elements G is sid to from group if in G there is defined inry opertion, clled the product nd denoted y such tht., G implies G (Closure property) 2.,, c G implies ( c) = ( ) c (ssocitive lw) 3. There exists n element e G such tht e = e = for ll G (the existence of identity element in G). 4. For every G there exists n element - G such tht - = - = e (the existence of inverse in G). A group G is elin or commuttive if for every, G =. A group, which is not elin, is clled non-elin. The numer of distinct elements in G is clled the order of G; denoted y ο (G) = G. If ο (G) is finite we sy G is of finite order otherwise G is sid to e of infinite order. DEFINITION..7: Let (G, ο) nd (G, ) e two groups. A mp φ: G to G is sid to e group homomorphism if φ ( ) = φ() φ() for ll, G. DEFINITION..8: Let (G, ) e group. A non-empty suset H of G is sid to e sugroup of G if (H, ) is group, tht is H itself is group. For more out groups refer. (I. N. Herstein nd M. Hll). Throughout this ook y S n we denote the set of ll one to one mppings of the set X = {x,, x n } to itself. The set S n together with the composition of mppings s n opertion forms non-commuttive group. This group will e ddressed in this ook s symmetric group of degree n or permuttion group on n elements. The order of S n is finite, only when n is finite. Further S n hs sugroup of order n!/2, which we denote y A n clled the lternting group of S n nd S = Z p \ {} when p is prime under the opertions of usul multipliction modulo p is commuttive group of order p-. Now we just recll the definition of Smrndche semigroup nd give some exmples. As this notion is very new we my recll some of the importnt properties out them. DEFINITION..9: The Smrndche semigroup (S-semigroup) is defined to e semigroup A such tht proper suset of A is group. (with respect to the sme induced opertion). DEFINITION..: Let S e S semigroup. If every proper suset of A in S, which is group is commuttive then we sy the S-semigroup S to e Smrndche commuttive semigroup nd if S is commuttive semigroup nd is S-semigroup then oviously S is Smrndche commuttive semigroup. 9
Let S e S-semigroup, ο(s) = numer of elements in S tht is the order of S, if ο(s) is finite we sy S is finite S-semigroup otherwise S is n infinite S-semigroup. Exmple..: Let Z 2 = {,, 2,, } e the modulo integers under multipliction mod 2. Z 2 is S-semigroup for the sets A = {, 5}, A 2 = {9, 3}, A 3 = {4, 8} nd A 4 = {, 5, 7, } re sugroups under multipliction modulo 2. Exmple..2: Let S(5) e the symmetric semigroup. S 5 the symmetric group of degree 5 is proper suset of S(5) which is group. Hence S(5) is Smrndche semigroup. Exmple..3: Let M n n = {( ij ) / ij Z} e the set of ll n n mtrices; under mtrix multipliction M n n is semigroup. But M n n is S-semigroup if we tke P n n the set of ll is non-singulr mtrixes of M n n, it is group under mtrix multipliction. Exmple..4: Z p = {,, 2,, p-} is semigroup under multipliction modulo p. The set A = {, p-} is sugroup of Z p. Hence Z p for ll primes p is S- semigroup. For more out S-semigroups one cn refer [9,,,8]. PROBLEMS:. For the semigroup S 3 3 = {( ij ) / ij Z 2 = {, }}; (the set of ll 3 3 mtirixes with entries from Z 2 ) under multipliction. i. Find the numer of elements in S 3 3. ii. Find ll the idels of S 3 3. iii. Find only the right idels of S 3 3. iv. Find ll susemigroups of S 3 3. 2. Let S(2) e the set of ll mppings of set X = {, 2,, 2} with 2 elements to itself. S(2) = S(X) is semigroup under composition of mppings. i. Find ll susemigroups of S(X) which re not idels. ii. Find ll left idels of S(X). iii. How mny two sided idels does S(X) contin? 3. Find ll the idels of Z 28 = {,, 2,, 27}, the semigroup under multipliction modulo 28. 4. Construct homomorphism etween the semigroups. S 3 3 given in prolem nd Z 28 given in prolem 3. Find the kernel of this homomorphism. (φ: S 3 3 Z 28 ) where ker φ = {x S 3 3 / φ(x) = }.
5. Does there exist n isomorphism etween the semigroups S(4) nd Z 256? Justify your nswer with resons. 6. Find ll the right idels of S(5). Cn S(5) hve idels of order 2? 7. Find ll the sugroups of S 4. 8. Does there exist n isomorphism etween the groups G = g/g 6 = nd S 3? 9. Cn you construct group homomorphism etween g = g/g 6 = nd S 4? Prove or disprove.. Does there exist group homomorphism etween G = g/g = nd the symmetric group S 5?. Cn we hve group homomorphism etween G = g/g p = nd the symmetric group S q (where p nd q re two distinct primes)? 2. Find group homomorphism etween D 2n nd S n. (D 2n is clled the dihedrl group of order 2n given y the following reltion, D 2n = {, / 2 = n = ; = }. 3. Give n exmple of S-semigroup of order 7 (other thn Z 7 ). 4. Does S-semigroup of order 2 exist? Justify! 5. Find S-semigroup of order 6. 6. Find ll sugroups of the S-semigroup, Z 24 = {,, 2,, 23} under multipliction modulo 24. 7. Cn Z 25 = {,, 2,, 24} hve suset of order 6 which is group (Z 25 is semigroup under usul multipliction modulo 25)? 8. Let Z 2 ={,, 2,, 2} e the semigroup under multipliction modulo 2. Cn Z 2 hve sugroups of even order? If so find ll of them..2 Lttices In this section we just recll the sic results out lttices used in this ook. DEFINITION.2.: Let A nd B e non-empty sets. A reltion R from A to B is suset of A B. Reltions from A to B re clled reltions on A, for short, if (, ) R then we write R nd sy tht ' is in reltion R to. Also if is not in reltion R to, we write R/. A reltion R on non-empty set A my hve some of the following properties: R is reflexive if for ll in A we hve R.
R is symmetric if for nd in A: R implies R. R is nti symmetric if for ll nd in A; R nd R imply =. R is trnsitive if for,, c in A; R nd Rc imply Rc. A reltion R on A is n equivlence reltion if R is reflexive, symmetric nd trnsitive. In the cse [] = { A R}, is clled the equivlence clss of for ny A. DEFINITION.2.2: A reltion R on set A is clled prtil order (reltion) if R is reflexive, nti symmetric nd trnsitive. In this cse (A, R) is prtilly ordered set or poset. We denote the prtil order reltion y or. DEFINITION.2.3: A prtil order reltion on A is clled totl order if for ech, A, either or. {A, } is clled chin or totlly ordered set. Exmple.2.: Let A = {, 2, 3, 4, 7}, (A, ) is totl order. Here is the usul less thn or equl to reltion. Exmple.2.2: Let X = {, 2, 3}, the power set of X is denoted y P(X) = {φ, X, {}, {}, {c}, {, }, {c, }, {, c}}. P(X) under the reltion inclusion s susets or continment reltion is prtil order on P(X). It is importnt or interesting to note tht finite prtilly ordered sets cn e represented y Hsse Digrms. Hsse digrm of the poset A given in exmple.2.: 7 4 3 2 Hsse digrm of the poset P(X) descried in exmple.2.2 is s follows: Figure.2. {,2,3} {, 2} {2, 3} {, 3} {2} {} {3} φ Figure 2.2.2
DEFINITION.2.4: Let (A, ) e poset nd B A. i) A is clled n upper ound of B if nd only if for ll B,. ii) A is clled lower ound of B if nd only if for ll B;. iii) The gretest mongst the lower ounds, whenever it exists is clled the infimum of B, nd is denoted y inf B. iv) The lest upper ound of B whenever it exists is clled the supremum of B nd is denoted y sup B. Now with these notions nd nottions we define semilttice. DEFINITION.2.5: A poset (L, ) is clled semilttice order if for every pir of elements x, y in L the sup (x, y) exists (or equivlently we cn sy inf (x, y) exist). DEFINITION.2.6: A poset (L, ) is clled lttice ordered if for every pir of elements x, y in L the sup (x, y) nd inf (x, y) exists. It is left for the reder to verify the following result. Result:. Every ordered set is lttice ordered. 2. In lttice ordered set (L, ) the following sttements re equivlent for ll x nd y in L.. x y. Sup (x, y) = y c. Inf (x, y) = x. Now s this text uses lso the lgeric opertions on lttice we define n lgeric lttice. DEFINITION.2.7: An lgeric lttice (L,, ) is non-empty set L with two inry opertions (join) nd (meet) (lso clled union or sum nd intersection or product respectively) which stisfy the following conditions for ll x, y, z L. L. x y = y x, x y = y x L 2. x (y z) = (x y) z, x (y z) = (x y) z L 3. x (x y) = x, x (x y) = x. Two pplictions of L 3 nmely x x = x (x (x x)) = x led to the dditionl condition L 4. x x = x, x x = x. L is the commuttive lw, L 2 is the ssocitive lw, L 3 is the sorption lw nd L 4 is the idempotent lw. The connection etween lttice ordered sets nd lgeric lttices is s follows: Result: 3
. Let (L, ) e lttice ordered set. If we define x y = inf (x, y) nd x y = sup (x, y) then (L,, ) is n lgeric lttice. 2. Let (L,, ) e n lgeric lttice. If we define x y if nd only if x y = x (or x y if nd only if x y = y) then (L, ) is lttice ordered set. This result is left s n exercise for the reder to verify. Thus it cn e verified tht the ove result yields one to one reltionship etween lgeric lttices nd lttice ordered sets. Therefore we shll use the term lttice for oth concepts. L = o(l) denotes the order (tht is crdinlity) of the lttice L. Exmple.2.3: 5 L 4 e lttice given y the following Hsse digrm: c This lttice will e clled s the pentgon lttice in this ook. Exmple.2.4: Let Figure.2.3 5 L 3 e the lttice given y the following Hsse digrm: c This lttice will e ddressed s dimond lttice in this ook. DEFINITION.2.8: Let (L, ) e lttice. If is totl order on L nd L is lttice order we cll L chin lttice. Thus we see in chin lttice L we hve for every pir, L we hve either or. Chin lttices will ply mjor role in this ook. Exmple.2.5: Let L e [, ] ny closed intervl on the rel line, [, ] under the totl order is chin lttice. Figure.2.4 4
Exmple.2.6: [, ) is lso chin lttice of infinite order. Left for the reder to verify. Exmple.2.7: [-, ] is chin lttice of infinite crdinlity. Exmple.2.8: Tke [, ] = L the two element set. L is the only 2 element lttice nd it is chin lttice hving the following Hsse digrm nd will e denoted y C 2. Figure.2.5 DEFINITION.2.9: A non-empty suset S of lttice L is clled sulttice of L if S is lttice with respect to the restriction of nd of L onto S. DEFINITION.2.: Let L nd M e ny two lttices. A mpping f: L M is clled. Join homomorphism if x y = z f(x) f(y) = f(z) 2. Meet homomorphism if x y = z f(x) f(y) = f(z) 3. Order homomorphism if x y imply f(x) f(y) for ll x, y L. f is lttice homomorphism if it is oth join nd meet homomorphism. Monomorphism, epimorphism, isomorphism of lttices re defined s in the cse of other lgeric structures. DEFINITION.2.: A lttice L is clled modulr if for ll x, y, z L, x z imply x (y z) = (x y) z. DEFINITION.2.2: A lttice L is clled distriutive if either of the following conditions hold good for ll x, y, z in L. x (y z) = (x y) (x z) or x (y z) = (x y) (x z) clled the distriutivity equtions. It is left for reder to verify the following result: Result: A lttice L is distriutive if nd only if for ll x, y, z L. (x y) (y z) (z x) = (x y) (y z) (z x). Exmple.2.9: The following lttice L given y the Hsse digrm is distriutive. c Figure.2.6 5
Exmple.2.: d c e This lttice is non-distriutive left for the reder to verify. Exmple.2.: Prove P(X) the power set of X where X = (, 2) is lttice with 4 elements given y the following Hsse digrm: Figure.2.7 X = {,2} {} {2} {φ} Figure.2.8 Exmple.2.2: The lttice e c d is modulr nd not distriutive. Left for the reder to verify. Figure.2.9 6
DEFINITION.2.3: A lttice L with nd is clled complemented if for ech x L there is tlest one element y such tht x y = nd x y =, y is clled complement of x. Exmple.2.3: The lttice with the following Hsse digrm: x x 2 x 3 x 4 Figure.2. is such tht x i x j =, i j; x i x j =, i j, ech x i hs complement x j, i j. Result: If L is distriutive lttice then ech x L hs tmost one complement which is denoted y x'. This is left for the reder to verify. DEFINITION.2.4: A complemented distriutive lttice is clled Boolen lger (or Boolen lttice). Distriutivity in Boolen lger gurntees the uniqueness of complements. DEFINITION.2.5: Let B nd B 2 e two Boolen lgers. The mpping φ: B B 2 is clled Boolen lger homomorphism if φ is lttice homomorphism nd for ll x B, we hve φ (x') = (φ(x))'. Exmple.2.4: Let X = {x, x 2, x 3, x 4 }. P(X) = power set of X, is Boolen lger with 6 elements in it. This is left for the reder to verify. PROBLEMS:. Prove the dimond lttice is non-distriutive ut modulr. 2. Prove the pentgon lttice is non-distriutive nd non-modulr. 3. Prove the lttice with Hsse digrm is non-modulr. e g f h Figure.2. 7
4. Find ll sulttices of the lttice given in Prolem 3. Does this lttice contin the pentgon lttice s sulttice? 5. Prove ll chin lttices re distriutive. 6. Prove ll lttices got from the power set of set is distriutive. 7. Prove lttice L is distriutive if nd only if for ll x, y, z L, x y = x z nd x y = x z imply y = z. 8. Prove for ny set X with n elements P(X), the power set of X is Boolen lger with 2 n elements in it. 9. Is the lttice with the following Hsse digrm, distriutive? complemented? modulr? d c. Prove for ny lttice L without using the principle of dulity the following conditions re equivlent..,, c L, ( ) c = ( c) ( c) 2.,, c L, ( ) c = ( c) ( c). Figure.2.2. Prove if,, c re elements of modulr lttice L with the property ( ) c = then ( c) =. 2. Prove in ny lttice we hve [(x y) (x z)] [(x y) (y z)] = x y for ll x, y, z L. 3. Prove lttice L is modulr if nd only if for ll x, y, z L, x (y (x z)) = (x y) (x z). 4. Prove hs 2 complements nd c in the pentgon lttice given y the digrm c 8
5. Give two exmples of lttices Figure of.2.3 order 8 nd 6, which re not Boolen lgers. 6. How mny Boolen lgers re there with four elements,, nd?.3 Rings nd Fields In this section we minly introduce the concept of ring nd field. This is done for two resons, one to enle one to compre field nd semifield. Second, to study group rings nd semigroup rings. We do not give ll the properties out field ut wht is essentil lone is given, s the ook ssumes tht the reder must hve good ckground of lger. DEFINITION.3.: A ring is set R together with two inry opertions + nd clled ddition nd multipliction, such tht. (R, +) is n elin group 2. The product r s of ny two elements r, s R is in R nd multipliction is ssocitive. 3. For ll r, s, t R, r (s + t) = r s + r t nd (r + s) t = r t + s t (distriutive lw). We denote the ring y (R, +, ) or simply y R. In generl the neutrl element in (R,+) will lwys e denoted y, the dditive inverse of r R is r. Insted of r s we will denote it y rs. Clerly these rings re "ssocitive rings". Let R e ring, R is sid to e commuttive, if = for ll, R. If there is n element R such tht r = r = r for ll r R, then is clled the identity (or unit) element. If r s = implies r = or s = for ll r, s R then R is clled integrl. A commuttive integrl ring with identity is clled n integrl domin. If R \ {} is group then R is clled skew field or division ring. If more over, R is commuttive we spek of field. The chrcteristic of R is the smllest nturl numer k with kr = r + + r (k times) equl to zero for ll r R. We then write k = chrcteristic R. If no such k exists we put chrcteristic R =. Exmple.3.: Q e the set of ll rtionls. (Q, +, ) is field of chrcteristic. Exmple.3.2: Let Z e the set of integers. (Z, +, ) is ring which is in fct n integrl domin. Exmple.3.3: Let M n n e the collection of ll n n mtrices with entries from Q. M n n with mtrix ddition nd mtrix multipliction is ring which is noncommuttive nd this ring hs zero divisors tht is, M n n is not skew field or division ring. 9
Exmple.3.4: Let M' n n denote the set of ll non-singulr mtrices with entries from Q tht is given in exmple.3.3. Clerly M' n n is division ring or skew field. Exmple.3.5: Let R e the set of rels, R is field of chrcteristic. Exmple.3.6: Let Z 28 = {,, 2,, 27}. Z 28 with usul ddition nd multipliction modulo 28 is ring. Clerly Z 28 is commuttive ring with 7.4 (mod 28) tht is Z 28 hs zero divisors. Exmple.3.7: Let Z 23 = {,, 2,, 22} e the ring of integers modulo 23. Z 23 is field of chrcteristic 23. DEFINITION.3.2: Let F e field. A proper suset S of F is sid to e sufield if S itself is field under the opertions of F. DEFINITION.3.3: Let F e field. If F hs no proper sufields then F is sid to e prime field. Exmple.3.8: Z p = {,, 2,, p-} where p is prime, is prime field of chrcteristic p. Exmple.3.9: Let Q e the field of rtionls. Q hs no proper sufield. Q is the prime field of chrcteristic ; ll prime fields of chrcteristic re isomorphic to Q. Exmple.3.: Let R e the field of rels. R hs the suset Q R nd Q is field; so R is not prime field nd chrcteristic R =. DEFINITION.3.4: Let R e ny ring. A proper suset S of R is sid to e suring of R if S is ring under the opertions of R. Exmple.3.: Let Z 2 = {,, 2,, 9} is the ring of integers modulo 2. Clerly A = {, } is suring of Z 2. Exmple.3.2: Let Z e the ring of integers 5Z Z is the suring of Z. Exmple.3.3: Let R e commuttive ring nd R[x] e the polynomil ring. R R[x] is suring of R[x]. In fct R[x] is n integrl domin if nd only if R is n integrl domin (left s n exercise for the reder to verify). DEFINITION.3.5: Let R nd S e ny two rings. A mp φ: R S is sid to e ring homomorphism if φ ( + ) = φ () + φ() nd φ () = φ() φ () for ll, R. DEFINITION.3.6: Let R e ring. I non-empty suset of R is clled right (left) idel of R if. I is suring. 2. For r R nd i I, ir I (or ri I). 2
If I is simultneously oth right nd left idel of R we sy I is n idel of R. Thus idels re surings ut ll surings re not idels. Exmple.3.4: Let Z e the ring of integers. pz = {, ±p, ±2p, } for ny p Z is n idel of Z. Exmple.3.5: Let Z 2 = {,, 2,, } e the ring of integers modulo 2. I = {, 6} is n idel of Z 2, P = {, 3, 6, 9} is lso n idel of Z 2 I 2 = {, 2, 4, 6, 8, } is n idel of Z 2. Exmple.3.6: Let R[x] e polynomil ring. p(x) = p + p x + + p n x n e polynomil of degree n (p n ). Clerly p(x) genertes n idel. We leve it for the reder to check this fct. We denote the idel generted y p(x) y p(x). DEFINITION.3.7: Let φ: R R' e ring homomorphism the kernel of φ denoted y ker φ = {x R / φ(x) = } is n idel of R. DEFINITION.3.8: Let R e ny ring, I n idel of R. The set R / I = { + I / R} is defined s the quotient ring. For this quotient ring, I serves s the dditive identity. The reder is requested to prove R / I is ring. PROBLEMS:. Let F e field. Prove F hs no idels. 2. Find ll idels of the ring Z 24. 3. Prove Z 29 hs no idels. M 2 2 3 2, M 2 2 is ring under usul c d mtrix ddition nd mtrix multipliction. i. Find one right idel of M 2 2. ii. Find one left idel of M 2 2. iii. Find n idel of M 2 2. 4. Let =,,c,d Z = {,, } 5. In prolem 4 find suring of M 2 2, which is not n idel of M 2 2. 6. Let Z 7 [x] e the polynomil ring. Suppose p(x) = x 2 + 3. Find the idel I generted y p(x). 7. Let Z 2 = {,, 2,, } e the ring of modulo integers 2. Let the idel I = {, 2, 4, 6, 8, }. Find the quotient ring Z 2 / I. Is Z 2 / I field? 8. Let Z 7 [x] e the polynomil ring over Z 7. I = x 3 + e the idel generted y the polynomil p(x) = x 3 +. Find Z 7 [x] / x 3 +. When will Z 7 [x] / x 3 + e field? 2
9. Find [ ] Z x < x + > 2.. Find ll principl idels in Z 5 3 [ x] = {ll polynomils of degree less thn or equl to 5}. (Hint: We sy ny idel is principl if it is generted y single element).. Construct prime field with 53 elements. 2. Prove Z 2 [x] / x 2 + x + is non-prime field with 4 elements in it. 3. Let Z e the ring of integers, prove nz for some positive integer n is principl idel of Z. 4. Cn Z hve idels, which re not principl? 5. Cn Z n (n ny positive integer) hve idels, which re not principl idels of Z n? 6. Let Z 24 = {,, 2,, 23} e the ring of integers modulo 24. Find n idel I in Z 24 so tht the quotient ring Z 24 / I hs the lest numer of elements in it..4 Vector spces In this section we introduce the concept of vector spces minly to compre nd contrst with semivector spces uilt over semifields. We just recll the most importnt definitions nd properties out vector spces. DEFINITION.4.: A vector spce (or liner spce) consists of the following. field F of sclrs. 2. set V of ojects clled vectors. 3. rule (or opertion) clled vector ddition, which ssocites with ech pir of vectors α, β in V vector α + β in V in such wy tht i. ddition is commuttive, α + β = β + α. ii. ddition is ssocitive; α + (β + γ) = (α + β) + γ. iii. there is unique vector in V, clled the zero vector, such tht α + = α for ll α V. iv. for ech vector α in V there is unique vector -α in V such tht α + (-α) =. v. rule (or opertion) clled sclr multipliction which ssocites with ech sclr c in F nd vector α in V vector cα in V clled the product of c nd α in V such tht..α = α for every α V.. (c, c 2 ) α = c (c 2 α). 22
c. c (α + β) = cα + cβ. d. (c + c 2 )α = c α + c 2 α for c, c 2, c F nd α, β V. It is importnt to stte tht vector spce is composite oject consisting of field F, set of ' vectors' nd two opertions with certin specil properties. The sme set of vectors my e prt of numer of distinct vector spces. When there is no chnce of confusion, we my simply refer to the vector spce s V. We shll sy 'V is vector spce over the field F'. Exmple.4.: Let R[x] e the polynomil ring where R is the field of rels. R[x] is vector spce over R. Exmple.4.2: Let Q e the field of rtionls nd R the field of rels. R is vector spce over Q. It is importnt nd interesting to note tht Q is not vector spce over R in the exmple.4.2. Exmple.4.3: Let F e ny field V = F F = {(, ) /, F}. It is left for the reder to verify V is vector spce over F. Exmple.4.4: Let V = {M n m } = {( ij ) / ij Q}. V is the set of ll n m mtrices with entries from Q. It is esily verified tht V is vector spce over Q. DEFINITION.4.2: Let V e vector spce over the field F. Let β e vector in V, β is sid to e liner comintion of vectors α,, α n in V provided there exists sclrs n c, c 2,, c n in F such tht β = c α + + c n α n = c iα i. DEFINITION.4.3: Let V e vector spce over the field F. A suspce of V is suset W of V which is itself vector spce over F with the opertions of vector ddition nd sclr multipliction on V. DEFINITION.4.4: Let S e set of vectors in vector spce V. The suspce spnned y S is defined to e the intersection W of ll suspces of V which contin S. When S is finite set of vectors sy S = {α,, α n } we shll simply cll W the suspce spnned y the vectors α, α 2,, α n. DEFINITION.4.5: Let V e vector spce over F. A suset S of V is sid to e linerly dependent (or simply dependent) if there exist distinct vectors α, α 2,, α n in S nd sclrs c,, c n in F not ll of which re such tht α c + + α n c n =. A set tht is not linerly dependent is clled linerly independent. If the set S contins only finitely mny vectors α, α 2,, α n we sometimes sy tht α,, α n re dependent (or independent) insted of sying S is dependent (or independent). DEFINITION.4.6: Let V e vector spce over the field F. A sis for V is linerly independent set of vectors in V, which spns the spce V. The spce V is finite i= 23
dimensionl if it hs finite sis which spns V, otherwise we sy V is infinite dimensionl. Exmple.4.5: Let V = F F F = {(x, x 2, x 3 ) / x, x 2, x 3 F} where F is field. V is vector spce over F. The set β = {(,, ), (,, ), (,, )} is sis for V. It is left for the reder to verify tht β spns V = F F F = F 3 ; we cn sy F 3 is vector spce over F of dimension three. Exmple.4.6: Let F e field nd F n = F F (n times), F n is vector spce over F. A set of sis for F n over F is β = {(,,,, ), (,,,, ), (,,,,, ),, (,,,, )}. It cn e shown, F n is spnned y β the dimension of F n is n. We cll this prticulr sis s the stndrd sis of F n. Exmple.4.7: Let F n [x] e vector spce over the field F; where F n [x] contins ll polynomils of degree less or equl to n. Now β = {, x, x 2,, x n } is sis of F n [x]. The dimension of F n [x] is n +. Exmple.4.8: Let F[x] e the polynomil ring which is vector spce over F. Now the set {, x, x 2,, x n, } is sis of F[x]. The dimension of the vector spce F[x] over F is infinite. Remrk: A vector spce V over F cn hve mny sis ut for tht vector spce the numer of elements in ech of the sis is the sme; which is the dimension of V. DEFINITION.4.7: Let V nd W e two vector spces defined over the sme field F. A liner trnsformtion T: V W is function from V to W such tht T (cα + β) = ct (α) + T(β) for ll α, β V nd for ll sclrs c F. Remrk: The liner trnsformtion leves the sclrs of the field invrint. Liner trnsformtion cnnot e defined if we tke vector spces over different fields. If W = V then the liner trnsformtion from V to V is clled the liner opertor. DEFINITION.4.8: Let L (V, W) denote the collection of ll liner trnsformtion of the vector spce V to W, V nd W vector spces defined over the field F. L(V, W) is vector spce over F. Exmple.4.9: Let R 3 e vector spce defined over the rels R. T(x, x 2, x 3 ) = (3x, x, x 2, 2x + x 2 + x 3 ) is liner opertor on R 3. Exmple.4.: Let R 3 nd R 2 e vector spces defined over R. T is liner trnsformtion from R 3 into R 2 given y T(x, x 2, x 3 ) = (x + x 2, 2x 3 -x ). It is left for the reder to verify T is liner trnsformtion. Exmple.4.: Let V = F F F e vector spce over F. Check whether the 3 sets re 3 distinct sets of sis for V.. {(, 5, ), (, 7, ), (3, 8, 8)}. 2. {(4, 2, ), (2,, 4), (, 4, 2)}. 24
3. {(-7, 2, ), (, -3, 5), (7,, -)}. PROBLEMS:. Let M 3 5 = {( ij ) ij Q} denote the set of ll 3 5 mtrices with entries from Q the rtionl field. i. Prove M 3 5 is vector spce over Q the rtionls. ii. Find sis of M 3 5. iii. Wht is the dimension of M 3 5? 2. Prove L(V, W) is vector spce over F if V nd W re vector spces over F. 3. Let V e vector spce of dimension 3 over field F nd W e vector spce of dimension 5 defined over F. Find the dimension of L(V, W) over F. 4. Suppose V is vector spce of dimension n over F. If B = {v,, v n }nd B' = {w,, w n } re two distinct sis of V. Find method y which one sis cn e represented in terms of the other (The Chnge of Bsis rule). 5. Show the spces M n n = {( ij ) ij Q} the set of n n mtrices with entries from Q is isomorphic with L(V, V) = {set of ll liner opertors from V to V}. (V is n-dimensionl vector spce over Q). 6. Prove we cn lwys get mtrix ssocited with ny liner opertor from finite dimensionl vector spce V to V. 7. Let T e the liner opertor on R 4. T(x, x 2, x 3, x 4 ) = (x + 3x 3 x 4, x 3 + 3x 4 x 2, 5x 2 x 4, x + x 2 + x 3 + x 4 ). i. Wht is the mtrix of T in the stndrd sis for R 4? ii. Wht is the mtrix of T in the sis {α, α 2, α 3, α 4 } where α = (,,, ) α 2 = (,, 3, 4) α 3 = (, 5,, 2) α 4 = (,,, )..5 Group rings nd semigroup rings In this section we introduce the notion of group rings nd semigroup rings; the min motivtion for introducing these concepts is tht in this ook we will define nlogously group semirings nd semigroup semirings where the rings re replced y semirings. Severl new properties not existing is the cse of group rings is found in the cse of group semirings. Throughout this section y the ring R we men either R is field or R is commuttive ring with. G cn e ny group ut we ssume the opertion on the group G is only multipliction. S is semigroup under multipliction. 25
DEFINITION.5.: Let R e ring nd G group the group ring RG of the group G over the ring R consists of ll finite forml sums of the form α g (i runs over finite numer) where α i R nd g i G stisfying the following conditions: n i= n i. α ig i = βig i α i = βi for i =, 2,,n. i= ii. n n n α ig i + βig i = ( α i + βi ) g i. i= i= i= iii. n n n α ig i βih i = γ km k where g i h j = m k nd γ k = α iβ i= i= i= iv. r i g i = g i r i for ll g i G nd r i R. v. n n r rig i = (rri ) g i for r R nd i i i= i= RG is n ssocitive ring with R s its dditive identity. Since I R we hve G = G RG nd R e = R RG, where e is the identity of the group G. If we replce the group G y the semigroup S with identity we get the semigroup ring RS in the plce of the group ring RG. Exmple.5.: Q e the field of rtionls nd G = g / g 2 = e the cyclic group. The group ring QG = { + g, Q nd g G} is commuttive ring of chrcteristic. Exmple.5.2: Let Z 8 = {,, 2,, 7} e the ring of integers modulo 8. S 3 e the symmetric group of degree 3. Z 8 S 3 is the group ring of S 3 over Z 8. Z 8 S 3 is noncommuttive ring of chrcteristic 8. Exmple.5.3: Let R e the rel field, S n the symmetric group of degree n. The group ring RS n is non-commuttive ring of chrcteristic. This is not skew field for RS n hs zero divisors. Exmple.5.4: Let Z 5 = {,, 2, 3, 4} e the prime field of chrcteristic 5. G = g g 2 = e the cyclic group of order 2. The group ring Z 5 G is commuttive ring with chrcteristic 5 nd hs zero divisors. PROBLEMS:. Let Q e the field of rtionls. G = S 8 e the symmetric group of degree 8. Find in the group ring QG = QS, right idel nd n idel. 2. S(6) e the symmetric semigroup. Let Z 6 = {,, 2,, 5} e the ring of integers modulo 6. Find in the semigroup ring Z 6 S(6). i. Idels. i i i j 26
ii. iii. iv. Right idels. Zero divisors. Surings which re not idels. 3. Let Z 5 S 3 e the group ring of the group S 3 over the prime field Z 5. Z 4 S 7 the group ring of the symmetric group S 7 over Z 4. i. Construct ring homomorphism φ from Z 5 S 3 to Z 4 S 7. ii. Find ker φ. iii. Find the quotient ring Z 5 S 3 / ker φ. 4. Let Z 2 S 3 nd Z 2 S(3) e the group ring nd the semigroup ring. Cn we construct homomorphism from Z 2 S 3 to Z 2 S(3)? 5. Find ll zero divisors in Z 3 S(3), the semigroup ring of the semigroup S(3) over the prime field Z 3. 6. Find ll zero divisors in Z 3 S 3, the group ring of the group S 3 over the ring Z 3. 7. Z 3 S(3) or Z 3 S 3 which hs more numer of zero divisors? (Hint: We know S 3 S(3) use this to prove the result). Supplementry Reding. Birkhoff, G. Lttice Theory. Americn Mthemticl Society, Providence, R.I., 967. 2. Birkhoff, G. nd Brtee, T.C. Modern Applied Alger. Mc-Grw Hill, New York, 97. 3. Grtzer, G. Lttice Theory. Freemn, Sn Frncisco, 97. 4. Hll, Mrshll. Theory of Groups. The Mcmilln Compny, New York, 96. 5. Herstein, I. N. Topics in Alger. 2 nd Ed. Wiley, New York, 975. 6. Lng, S. Alger. Addison-Wesley, 967. 7. Ngt, M. Field Theory. Mrcel Dekker, NewYork-Bsel, 977. 8. Pssmn, D.S. Infinite Group Rings. Mrcel Dekker, New York-Bsel, 97. 9. Pdill, R. Smrndche Algeric Structures. Smrndche Notions Journl, USA, Vol.9, No. -2, 36-38, 998.. Pdill, R. Smrndche Algeric Structures. Bulletin of Pure nd Applied Sciences, Delhi, Vol. 7 E, No., 9-2, 998. http://www.gllup.unm.edu/~smrndche/alg-s-txt.txt 27
. Smrndche, F. Specil Algeric Structures. Collected Ppers, Vol. III, Add, Orde, 78-8, 2. 2. Vsnth Kndsmy, W.B. On zero divisors in reduced group rings of ordered groups. Proc. of the Jpn Acdemy, Vol. 6, 333-334, 984. 3. Vsnth Kndsmy, W.B. Semi-idempotents in semigroup rings. Journl of Guizhou Inst. of Tech., Vol. 8, 73 74, 989. 4. Vsnth Kndsmy, W.B. Idempotents in the group ring of cyclic group. Vikrm Mth. Journl, Vol. X, 59-73, 99. 5. Vsnth Kndsmy, W.B. Filil semigroups nd semigroup rings. Liertrs Mthemtic, Vol. 2, 35-37, 992. 6. Vsnth Kndsmy, W.B. On strictly right chin group rings. Hunn Annele Mths. Vol. 4, 47-99, 994. 7. Vsnth Kndsmy, W. B. Smrndche Semirings nd Semifields. Smrndche Notions Journl, Vol. 7, -2-3, 88-9, 2. http://www.gllup.unm.edu/~smrndche/semirings.pdf 8. Vsnth Kndsmy, W. B. Smrndche Semigroups, Americn Reserch Press, Rehooth, 22. http://www.gllup.unm.edu/~smrndche/vsnth-book.pdf 28
CHAPTER TWO SEMIRINGS AND ITS PROPERTIES The study of the concept of semiring is very megre. In my opinion I hve not come cross textook tht covers completely ll the properties of semirings. Hence this complete chpter is devoted to introduction of semirings, polynomil semirings nd mny new properties out it, nlogous to rings. This chpter lso gives for the ske of completeness the definition of severl types of specil semirings like -semirings, congruence simple semirings nd so on. We do not intend to give ll definition or ll properties insted we expect the reder to refer those ppers which re enlisted in the supplementry reding t the end of this chpter. This chpter hs six sections. In sections one nd two we define semirings nd give exmples nd prove some sic properties. Section three shows how lttices re used to construct semirings. Polynomil semirings re introduced in section four. Section five defines nd reclls the definitions of group semirings nd semigroup semirings. The finl section minly reclls some of the specil types of semirings like c-semirings, -semirings, inductive -semirings etc. 2. Definition nd exmples of semirings In this section we introduce the concept of semirings nd give some exmples. This is minly crried out ecuse we do not hve mny textook for semirings except in the ook 'Hndook of Alger' Vol. I, y Udo, which crries section on semirings nd semifields. DEFINITION (LOUIS DALE): Let S e non-empty set on which is defined two inry opertions ddition '+' nd multipliction ' ' stisfying the following conditions:. (S, +) is commuttive monoid. 2. (S, ) is semigroup. 3. ( + ) c = c + c nd ( + c) = + c for ll,, c in S. Tht is multipliction ' ' distriutes over the opertion ddition '+'. (S, +, ) is semiring. DEFINITION (LOUIS DALE): The semiring (S, +, ) is sid to e commuttive semiring if the semigroup (S, ) is commuttive semigroup. If (S, ) is not commuttive semigroup we sy S is non-commuttive semiring. DEFINITION (LOUIS DALE): If in the semiring (S, +, ), (S, ) is monoid tht is there exists S such tht = = for ll S. We sy the semiring is semiring with unit. Throughout this ook Z + will denote the set of ll positive integers nd Z o = Z + {} will denote the set of ll positive integers with zero. Similrly Q o = Q + {} will 29
denote the set of ll positive rtionls with zero nd R o = R + {} denotes the set of ll positive rels with zero. DEFINITION (LOUIS DALE): Let (S, +, ) e semiring. We sy the semiring is of chrcteristic m if ms = s + + s (m times) equl to zero for ll s S. If no such m exists we sy the chrcteristic of the semiring S is nd denote it s chrcteristic S =. In cse S hs chrcteristic m then we sy chrcteristic S = m. Here it is interesting to note tht certin semirings hve no chrcteristic ssocited with it. This is the min devition from the nture of rings. We sy the semiring S is finite if the numer of elements in S is finite nd is denoted y S or o(s). If the numer of elements in S is not finite we sy S is of infinite crdinlity. Now we give some exmples of semirings. Exmple 2..: Let Z o = Z + {}. (Z o, +, ) is semiring of infinite crdinlity nd the chrcteristic Z o is. Further Z o is commuttive semiring with unit. Exmple 2..2: Let Q o = Q + {}. (Q o, +, o) is lso commuttive semiring with unit of infinite crdinlity nd chrcteristic Q o is. Exmple 2..3: Let M 2 2 = c / d,,c,d Z o = set of ll 2 2 mtrices with entries from Z o. Clerly (M 2 2, +, o) is semiring under mtrix ddition nd mtrix multipliction. M 2 2 is non-commuttive semiring of chrcteristic zero with unit element nd is of infinite crdinlity. Exmple 2..4: Let S e the chin lttice given y the following Hsse digrm: 9 7 5 3 S is semiring with inf nd sup s inry opertions on it. This is commuttive semiring of finite crdinlity or order. This semiring hs no chrcteristic ssocited with it. Figure 2.. 3
Exmple 2..5: Consider the following lttice given y the Hsse digrm: c d f e Figure 2..2 It cn e verified tht this lttice is lso semiring which is commuttive with finite crdinlity nd hs no chrcteristic ssocited with it. DEFINITION 2..: Let S nd S 2 e two semirings. The direct product of S S 2 = {(s, s 2 )/ s S nd s 2 S 2 } is lso semiring with component-wise opertion. Similrly if S, S 2, S n e n semirings. The direct product of these semirings denoted y S S 2 S 3 S n = {(s, s 2,, s n )/ s i S ; i =, 2,, n} is semiring lso known s the direct product of semirings. Exmple 2..6: Let Z o e the semiring Z o Z o Z o = {(,, c)/,, c Z o } is semiring. This enjoys vividly different properties from Z o. Exmple 2..7: Let S e the two-element chin lttice nd S 2 e the lttice given y the following Hsse digrm: S S 2 = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} is semiring under the opertions of the lttices, S S 2 hs the following Hsse digrm: Figure 2..3 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Figure 2..4 3
32 S = S S 2 is semiring (left for the reder s n exercise to verify). Exmple 2..8: Let S e the lttice with the following Hsse digrm S is commuttive semiring with unit nd is of finite crdinlity. Let S 2 2 = S x,,x x, x x x x x 4 3 2 4 3 2 = set of ll 2 2 mtrices with entries from the semiring S = {,,, }. Let A, B S 2 2, where A = 4 3 2 nd B = 4 3 2. Define '+' on S 2 2 s A + B = 4 3 2 + 4 3 2 = 4 4 3 3 2 2. Clerly (S 2 2, '+') is commuttive monoid. The mtrix cts s the dditive identity. For A, B S 2 2 define s A B = 4 3 2 4 3 2 = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 4 2 3 3 4 3 4 2 2 3 2. Clerly (S 2 2, ) is semigroup. It is left for the reder to verify (S 2 2, +, ) is semiring. This semiring is of finite crdinlity ut is non-commuttive for if A = nd B =. A B =. Now B A =. So A B B A for A, B S 2 2. Thus we hve seen semirings of chrcteristic which is oth commuttive nd noncommuttive hving infinite crdinlity. We re yet to find semiring of chrcteristic n, n nd we hve seen finite crdinlity semirings with no chrcteristic ssocited with it oth commuttive nd non-commuttive. PROBLEMS: Figure 2..5
. Give n exmple of semiring of order 27. 2. How mny elements does the semiring M 3 3 = {( ij )/ ij L where the lttice L is given y the following Hsse digrm}? c d 3. Prove M 3 3 given in Prolem 2 is non-commuttive semiring. 4. Find the unit element of M 3 3. 5. Let X = {x, x 2, x 3, x 4, x 5, x 6 } nd P(X) e the power set of X. Prove P(X) is semiring. 6. Cn non-commuttive semirings of finite order n exist for every integer n? 7. Cn ny finite semiring hve chrcteristic p? (p ny finite integer) Figure 2..6 8. Give n exmple of finite semiring which hs no chrcteristic ssocited with it. 9. Give n exmple of n infinite non-commuttive semiring of chrcteristic zero. 2.2 Semirings nd its properties In this section we introduce properties like susemirings, idels in semirings, zero divisors, idempotents nd units in semirings. DEFINITION 2.2.: Let S e semiring. P suset of S. P is sid to e susemiring of S if P itself is semiring. Exmple 2.2.: Let Z o e the semiring. 2Z o = {, 2, 4, } is susemiring of Z o. Exmple 2.2.2: Let Z o [x] e the polynomil semiring. Z o Z o [x] is susemiring of Z o [x]. 33
Exmple 2.2.3: Let S e distriutive lttice which is semiring given y the following Hsse digrm: c d e f g h Figure 2.2. L = {,, c, d, e, g, h, } is susemiring of S. DEFINITION 2.2.2: Let S e semiring. I e non-empty suset of S. I is right (left) idel of S if. I is susemiring. 2. For ll i I nd s S we hve is I (si I). DEFINITION 2.2.3: Let S e semiring. A non-empty suset I of S is sid to e n idel of S if I is simultneously right nd left idel of S. Exmple 2.2.4: Let Z o e the semiring. nz o for ny integer. n is n idel of Z o. Exmple 2.2.5: Let S e semiring given y the following lttice whose Hsse digrm : c d Figure 2.2.2 34
Clerly I = {, d, c,, } is n idel of S. DEFINITION 2.2.4: Let S e semiring. x S \ {} is sid to e zero divisor of S if there exists y in S such tht x y =. DEFINITION 2.2.5: Let S e non-commuttive semiring with unit. x S is sid to e hve right (left) unit if their exists y S such tht xy = (or yx = ). DEFINITION 2.2.6: Let S e semiring. x S is n idempotent, if x x = x 2 = x. Exmple 2.2.6: Let S e semiring given y the following Hsse digrm, S is such tht = i.e. is zero divisor nd 2 = nd 2 = so the semiring hs idempotents. Exmple 2.2.7: Let S e semiring given y S = Z o Z o Z o. S hs zero divisors given y = (8, 2) nd = (, 6, ) nd = (,, ). Exmple 2.2.8: Let S = Q o Q o Q o Q o e the semiring under component wise ddition nd multipliction. For S is semiring with unit (,,, ). Let = (3, ¼, 5/3, 7/2) S. The inverse of is = (/3, 4, 3/5, 2/7) S is such tht = (,,, ). Thus S hs units. DEFINITION 2.2.7: Let S nd S' e two semirings. A mpping φ : S S' is clled the semiring homomorphism if φ( + ) = φ() + φ() nd φ( ) = φ() φ() for ll, S. If φ is one to one nd onto mp we sy φ to e semiring isomorphism. DEFINITION (LOUIS DALE): Let S e semiring. We sy S is strict semiring if + = implies = nd =. Chris Monico clls this concept s zero sum free. Exmple 2.2.9: Z o the semiring is strict semiring. Exmple 2.2.: The semiring Q o is strict semiring. Figure 2.2.3 35