Kyungpook "'I.th. J. Volume 22‘ Numbcc 2 December. 1982
THE THEORY OF SEMIFIELDS
By Sidney S. Mitchell and Porntip Sinutoke
1. Introduction
The first mathematical object studied Îs thc non-negative intcgers. Thc sc‘ onι mathematical object studied is the non-ncgatÎ\'C rationals. lt is only later on that one studies the integers Z. the rationals Q and the rca!s R. The non !'JcgatÎ\-e integers. Z and Q are vcry important examples of a semiring, ring and field respectively. Ring theory and field theory arc t\VO of thc most ÎmportÐ.i!í.. and most developed theories in algehra. Scmiring theory has been s! 。 、，vCr to develope but is now beginning [0 attract thc attention of algebra more and mQrc. However the stud y of albegraic structures satisfying thc propertics of the non. negative rationals has been almost complctely ignored. Recently in [1] Dalc and Pitts ga\'e the definition of semifield. which is an algcbraic struCture satisfying the properties of the non.negative rationals but proved no theorems. ln [2] and [3] Stone defined the terms halffield and division halffield which are somewhat like the semi fields of Dale and Pitts but the operations are not assumed to be commutative among other differences. The purpose of this paper is to sct the foundations of the theory of semifields of Dale and Pitts so we shall considcr some of their most basic and elementarl' properues. We restrict ou rselves to the commutative case only. Hence in this paper the words fing and semiring 、vill mean commutative ring and commutative semiring. Since there are slight differences in the definitions of semirings in the literature we shall now give the one that we shall use throughout this paper.
DEFI:-IlTIO:-I. A semirillg is a triple (S. + .. ) where S is a nonempty set and + and . are binary operations on S such that (S. +) and (S•. ) are commutative semigroups and x(y+z)=xy+xz for all x.y.zES.
ln this paper we shall use the following notation:
+ ~+ 1) Z + = (1I EZ!n>O) 2) Zci =Z+U (1이 3) Z∞ = ru(∞ l 326 Sid Jley S. Milchell Q l1d POYll tip SiJmtoke
4) Q+ = [ rεQ ! r>O ) 5) Q;=Q+U IOl 6) Q￡= Q + Ul∞}
7) R + = [xE R !x > 0) 8) R;= R ~ U{O ) and 9) R:;' =R + IJ[∞ ) . All these sets with their usual addition and multiplication are semirings. A semiring with multiplicative zc rQ is a semiring S posscssing a distinguished oe lement aE S such that x'a=a fo r all xE S.
A semiring ,,,,-ith zero is a semiring S with multiplicative zero OES such that x+O=x fo r all x εS . We shall a lways denote the zero element of a semiring with zero by O. A semiring with infinity is a semiring S with mu1tiplicative zero ∞E S such
--r hat x+ ∞=∞ for all xE S. \Ve shall always denote the zero element of a semiring with infinity by ∞ . Note that 2) . 5) and 8) abo,.e arC semirings with zero and 3), 6) , 9) are ~emirings wi th infinity. A semiring w i 대 multipli cative zero may be neither a semiring 、vith ZC rO nQr a semiring with infinity. For example the direct proJuct semiring S= .z:; XZ ;;;' has (0, ∞) as a mu‘u삐j j i nf i ni냐ty of S. We shall see later that this cannot happen for semifields. \Ve shall now give our definition of a semifield which is slightly different [ro:TI tha t in [IJ.
DEFI:.J ITION. A semifield is a semi ring (K ‘ +, .) such that (K , .) is an abelian .g roup with zerQ i. e. there exists a distinguished element aEK such that (K- [a]. . ) is an abelian group and a.x= x.a=a for all. We shall always denote the multiplicati 、 e identity of a semi[ield br 1. E,.ery field is a semifield and the semi’ ings 2) .3).5), 6). 8) and 9) above are ,all examples of semifields whi ch are not fi elds. We shall now show that there .a re t ，‘ 。 ma jo r type3 of semifields. 2. O.Semifields and ∞ - semifields Let K be a semifield and let a denote the zero of K . Then ax=a for all xε K . ]‘he first natural question to ask is what is a +x? T HEOREM 1. Let K be a semifield alld a tlze 2ero elemel!t of K . T he eil Jz er .a .x=x for all x 드 K or a +x=a for all x든 K. The TheJry 01 Semtfields 327 PROOF. First we 5hall 5how that a+a=a. Let y=a+a. Then a=ay=a(a+a)= a-? +a-;;::a ? +a. CASE 1. There exist5 a nonzero "εK such that ,,+a=a. Let y be an arbítrary ‘ -‘ element in K. Then yx-'(,,+a) =yx ‘ a 80 y+a ;;:: a. Since y is arbitrary y+a-=a for all yEK 50 we are done. CASE 2. For all nonzero .tEK ,,+ a;ta. Then 1 +a 잊 a . Let μ= 1 +a. Then u. # a and u+a=( I + a )+a= I +(a+a)= I +a= κ . Hence there exists a nonzero uE K such that “ + a= ι. Let v be an arbítrary element of K. Then ν ，， -I( ，， + a)= (0“一 I)U 50 0 + a =u. Since ν is arbitrary v+a=v for all 1ε K 80 we are Jone. As a result of thís theorem we see that semílíeld5 are 01 two types. Eíthe r a+x=.t lor all x or a+ x=a for all x. In the fírst type. a behaves as an addíti\'e identity (whích is usually denoted by 0) so we call the fírst type 01 sermjicld a semi{ield o{ 2ero type or a O-semi{ield and we sball al“.ra ys denote the zero element of thís type of semifíeld by O. In tbe second type, a beha 、 es as an additive zero (whích is usually denoted by ∞ ) 50 we shall call the second type of semí fíeld a semi{ield o{ in{i1씨'y type or aη ∞ sem,jield and we shall always denote the ze ro element of tbís type of semifíeld by ∞ . lf we don't care wbich type 01 semifíeld we are 5tud yíng we shall just use the word semífíeld. Note that every fícld ís a O-semífield. Q,j and R; are O-semifields which a re not fields and Q! and R! a re 0。一 sem i fields. ln the proof 01 Theorem 1 we proved that if there exísts a nonzero " such lhat x+a=a then every nonzero element has this property and if there exists a nonzero x such that x+a=x then every non::ero element has this pl 이Jerty . The proof uses the fac t that e 、 ery nonzero element has a multiplicati\'e inverse. This type of 5tatement appear5 frequently in semifield theory i.e, if one nonzero eìement has a certain property then every nonzero element has this property. We shall ca\l this the principle o{ uni{ormity for semi fields. We sha\l soon see several other examples of the principle of unifo rmity for semifields. Let ( G, . ) be an abelian group witb zero, the zerO element denoted by ∞ (50 x. ∞ =∞ fo r a\l x εG ) . Define a binary operation + on G by ，， +y= ∞ for all x ， y εG . Then (G, +, .) is an ∞ semifield which is called the Irivial senujield. Let (G,. ) be an abelian group with ze ro, the zero element denoted by ∞. Define a b i 때 y 0 야 ration + on G by x +y= (말 ;f X 겸. Then (G, +, .) is an 328 Sidney S. MiJ chell alld Porlltip Sillutoke ∞ - sem ifie ld which is callcd the alm051 Irivial semifield. As a result. \IlC see that ∞ semifields are very pl entif 띠. In fact. e 、 ery abelian grou p with zcro can bc made into an ∞ -sc mifield in at least two difîerent ways. Let B= IO. l} and let . and + bc defined by 」으上 괴으上 0 1 0 1 1 I 1 1 Then (B, + .. ) is a fi ni te Q-semifield which is not a fi eld. We shall cal1 this semi iicld the Booleall semifield. This name was chosen for the following reaSQD. t et X = {x} bc a si ngleton set and let P(X ) denote the j:ower set of X 50 P(X )= 띠 . X ). Then P( X ) with thc binary operations of un ion and intersection is a semiri ng which is isomorphic tO B. The only examples 01 finite O-semifields that we have given so far arc the fi nite fields and the BooJean semifieJd. We shal1 show in the next section (Theorem 5) that these are in fact the onJy finite O-semifieJds tha t exist. Hence O.sem iiields are not so plentif비. The Boolean semifield is quite useful in set theory as the following result ShOW5. DEFI1\ITIO:-l. A semiring (S. u. n ) is said to bc a Booleo" s.miring if and onJy if 1) xnx= x=xux for a11 xES and 2) xu(yn ι ) = ( xuy ) n (x uz) for all x.y. zE S. Let Y be a nonempty set. Then ( P(Y). u. n) is a Boolean semiring ca11ed the 1,,1/ Boolean semirilzg 01 sπ "sels 01 Y. The semiring P (Y ) is isomorphic to a direct product of the Boolean semifield B as the fo11owing argument shows. Let Z =BY i. e. Z is the set 01 a11 maps from Y to B. Define a map 1: P(Y) -• Z as fo llows : if AEP(Y) Jet I (A) be the map ψ :Y-• B where ψ (y)= %% %혈 . Then clearly 1 is a se밍ml씨)lrJ띠lr띠rmgH 1 잃8omor S잉m따lκc야e a o- s히e ml“띠f“le터ld 잉i s the only typ야e 이o fs앞eml“떠fie잉l뼈d con따l“t떠a ining all add비lπ띠…tiive identity. a O-semifield is the only type of semifield in which the concept of an additive inverse of an element makes sensc. 1n an ∞- se mifield the additive behavior of ∞ is the same as the behavior of the set X in P (X) (the power set of X ) with respect to union. Hence we can use the language of set theory to define the following concept in semiring theory. Tlle Theory 01 Sem,jie/ds 329 DEFINITIO:-I. Let S be a semiring with infinity and let yES. Then zε5 is said to be a c0 11l ple11l.,,1 01 y if and only if y + z = ∞. Clearly ∞ is a complemcn[ 。 f every element of S and every element is a complement of ∞. DEFINIT IO N. Let S be a semiring with infinity and let xES. Then x is ， ai 이 to be limiled if and only if the only complement of " is ∞. If every γεS \ { ∞ l is limited then S is said to be limiled. We shall now show that if one nonzero element of a Q'semifield has a n additive inverse then all nonzero elements have an additive inverse. So again we have the principle of uniformity for semifield s. THEOR EM 2. Lel K be a Q'semilield. Then either every .tOnzero elen’‘”’t 01 K has an add,'tive ùzverse (i1l which case K t'S a field) Or no n01lzero element 01 K has an additfve inverse. PROOF. Suppose that the nonzero element x has an additive invcrsc J'. T hen X 十 y=Q. Let z be an arbitrary element of K. Then zx• l(x + y)=zx lO therelcre z+ zx ly = 0 ∞ z has an additive inverse. Since zEK is arbitrary, e,-ery zEK has an additive inverse. We shall nO \V show that there is also a principle of uniformity for complements III an ∞- semifield (Prof- osition 2 and Corollary 1 below) . First we‘ II need a preliminary definition. DEFINITlON. Let J{ bc an ∞一 sem i f i eld and let xEK. Then thc core 01 x. denoted by Cor(x) = (yεK l x + y = ∞ 1 . Clearly Cor(∞)= K. The following prop ~s i ti o n is easily prO\'ed. PROPOSITIO:\ 1. Lel K be an ∞-sell llfield . Then 1) ∞εCor ( ，， ) l or 011 x εκ 2) f o, 011 xεK. yECOI'(x) and zEK implies that y - zE Cor(,,) 3) f or 011 x. yEK\ {∞ ) ， yECor(x) i11ψl!es that y- lgCcr(x- l) 4) for all xECor( l ) aud ; EK, x +y + xyECor(l ) alld 5) for all xEK\ { ∞ x ' Cor(l ) =Cor(x), Part 5) of Proposition 1 immediatcly givcs us the following rcsu1ts. PIWPOSITlOK 2. Let K be an ∞ se mz꺼 e / d alld X , Y 1101Ziκifinity e/e11lcnts 01 K . Then the cardùzality 01 Cor(x) equals tlz e cordillalily of Cor(y), Í1‘l act , eαcι one ~ S a 1J‘ulliplicative translale 01 tI:e other. COROLLAR Y 1. 11 Olle notlù'lit,ity .Iem.nt 01 a’‘ ∞-semzfield K is Umiled t/len all n'J uinfinily elements 01 K are limiled. 330 Sidlley S. Mitchell and Porntip Sillutoke COR OLLARY 2. Lel K be a selllljield, Iet a detlote tlze zero 0/ K alld lel A= {xEK:x+y=a/or all yEK}. Th eη eitker A =tþ or A= {a} or A=K. We sha 11 give one more example 01 a principle 01 uniformity. DEFJ NlTJON. Let 5 be a semiring and .• an element of 5, Then x is said t。 be addilively callcellative (m씨tiplica tively cm,cellalive) if and only if for all y, zE 5, x +y=x+z ( XY=X2 ) implies that )'=2. From now on we sha11 abbreviate the words addith'ely canceIlative (multiplicatively canceIla tive) by A. C. (M. C. ) DEFI:-.1 ITIO:-i. A semiring 5 is said to be A. C, (M. C.) if and only if every X든 5 i5 A. C. (M. C. ). If a se miring 5 contains an additive identity 0 then 0 is A , C. and if 5 contains a multiplicative identity 1 then 1 is M. C. Also, if the ordcr of 5 is > 1 and 5 contains an additive ùlfini’ty ∞(i， e. x+ ∞=∞ 1" or a 11 xE5) then ∞ is not A, C. so 5 is not A. C. However we can give the fo II owing weaker definition of A. C. in thi5 case. D E FI ~ IT IO:\ . Let S be a semiring with additive infinity. Then 5 is said to be …ifillμ'y addilively callcelIalive if and only if for aIl x, y, zES, x +y=x+z and X"" ∞ implies tbat y=z. We shaIl abbreviate infinity additive canceIlativity by ∞ - A.C ， If a semiring S is of order >1 and contains a multiplicative 2ero 0 then 0 is not M. C. 50 S is not M, C, However we can gi 、 c the fo Ilowing 、λ’ ea ker definition of M. C, in this case. DEFIl\ITIO:\. Let S be a semiring with multiplicative zero. Then S is said to be zero 1II11ltiplicatively Ca/κ ellalive if and only if fo r aIl x, y, ιES x)' = xz and x""O implies that y=z. \Ve shall abbre 、 iate zerO multiplicative canceIlativity by o-M. C. The 5et 01 A. C. elements of a semiring S is either empty Or an additive s u bseπ ， i g r o up of S and the set 01 M. C, elements of S is eithcr empty or a multipli cative subsemigroup of S. The 5et 01 A. C. elements 0 1" a C-semifield K is an ideal of K and the set 01 A. C. elements C of an ∞ .semifie ld K has the properry that if xEC and yEK\ {∞ } then xyEC. Furthermore a semi field can be A. C, only if it is a O-semifjeld and in this case it always contains at least one A. C, element, 0, Thc be5t that an ∞ semifield can bc PROPOSITlO:-l 3. Let K be a semi지 eld. The1l ‘:'f Ol1 e notlzero element 0/ K is A. C. the1l all 1l011 zero elom.,.ts aro A.C. (so if K is a O-somifiold 11.011 K is A.C. aI.d '1 K is m. ∞-semtfield tllell K is ∞ A.C.) We shall close this section with lWO remarks. First, note that if J( is an ∞→ A. C . ∞ - semifie ld then K must be limitcd since if there exist x ， y E K\{∞ } such that x+y=oo then X T y = ∞ = x +∞ and x ， y "< ∞ so we get a contradiction. Secondly note that if K is a scmifield then K x K can never be a semifield (0, 0)= (1 , 0) since (0,1) and ( 1, 0)"<(0, 0), (0, 1) "<(0, 0). 3. Positive rational domains REMARK. Let K be a O-semifield which it not a field. Then by Theorem 2 if x, yE K are nonzero then x+y,. REMA RK. Let K be a limited ∞- semifie ld and let x, y EK\{∞) . Then ，， + y ，.< ∞. so again we get that (K\l∞ )， + , .) is a semiring 、.' ith the property that (K\ {에 , . ) is an abelian group. A se miring with this special property is very important in semifield theory and we shall call it a positive rational domain DEFE\lTIO:-l. A semiring (D. + , .) is said to be a þositiz'o raliollal domaill (PRD) if and only if ( D, .) is an abelian group. \Ve use the tcrm positive rational domain since the positive rationals are a model for this type of semiring. There is a natural bi jection between the PRD’ s and the O-semifields which arc not field s defincd as follows: gven a PRD D let 0 be a symbol not denoting any element of D. Extend the operations + and . on D to DU (0) by defining "+0=0+ ,, = ,, and ",0=0,_< = 0 for all xEDU (0). Then DU (0) is a semifield of 0 type which is not a Jïeld. By the fi rst remark in this section this correspondence is a bijection. Also. in the samc way there is a natural bijection between the PRD’s and the limited ∞ scmifields(in this case we define ，， +∞=∞ +x=x and x' ∞= ∞ 'x = ∞ for all "εD U ( ∞) ). These two observations show that the theory of PRD’ s plays an impo rtant role in the theory of semifields. Whenever discussing PRD’s we shaIl denote the multiplicativc identity by 1. The PRD’ s determine the best bchaved semifields. EXA ;lIPLES of PRD's 1) R+ with the usual addition and multiplicatio' is a PRD. 2) R • with the lIsual addition and ffiu ltipJication is a PRD. 332 Sidney S. Mitchell and Porll“p 5inι toke 3) {1} is a trivial example of a finite PRD. Note that a field or a semifield can never be a PRD since the zero element has no inverse. AIso, if D, and Dz are PRD’ s then D,X Dz with the product semiring structure is a PRD. Also, the principle of uniformity is true for PRD’s În a very strong form. namely if onc e]eIDcnt has a certain property then all elements ha 、 e this property (since cvery element has a multiplicative inverse). As a result, in a PRD if one element is then all elements are A. C. so the PRD is A.C. The trivial example of a finite P. R. D. given ahove is in fact the only example of a finite P. R. D. as the following theorem shows. THEOREM 4, There is 110 finite p , R , D. of order > 1. PROOF , Suppose that there exists a finite P. R. D. D of order n>1. We claim that for all zED\ {1} z+17즈 1. Suppose this claim is not true. then there exists an xED\{1} such that ,<+1=1. Let "'> 1 be the order of x in the group ( D, '). By using mathematical induction one can easily show that / + 1=1 for all , m-l kEZ ι. Hence Xlll ‘ + 1= 1. Since x(x"- ' + 1) =x, 1+ .<=.< and 50 .<=1 which is a cr:m traòiction. Thus the c1 aiID is true. 1f1JlεZ + define >>1 1=1+1+" , + 1(1/1 times). Then sincc D is finite, there exist m. kεZ + 5uch that m This is a \'ery important theorem in hoth semiring and semifield theory and it will be used se‘ eral times in this paper. The following results are interesting immediate applications of the theorem. COROLLA IIY. Jf 5 is a finite semiring of 0γder > 1, then 5 cannot be M , C. PROOF. If 5 is a finite M. C. semiring then (5, ,) is a finite cancellative commutati\'e se migroup 50 by a wcll knowD result in semigroup theory (5, .) is an abelian group. Hence 5 is a finite PRD. By the ahove theorem the order of 5 must be 1. 50 we see that finite cancellative semirings are very restricted. If the semiring .s is A. C. then the same proof as that given in the corollary above sho\\'s that Tlze Theory 01 Semi!ield s 333 S is a ring and if S is M. C. it must have order 1. Since a fi ni te A. C. semiring 15 a ring 、v e get that a finite A. C. se mifield mU5t be a field. THEOREM 5. Let K be a f inite semif ield. 1) [f K is a O'semif ield then K is eifher a f ield or Ihe Booleall semifield. 2) If K is a limifed ∞ - semifield thell K tnκst be Ihe allllost triνial ∞-se lJl ljiel d witl, two elemellts. (Hence ,j K is a finile ∞ - A.C. ∞ - semifield the7/ K is the .alnwst trivial ∞ - sem ，jield of order 2). PROOF. 1) Let K be a fi nite Q.semifield which is not a fi eld. Then K \ [이 15 .a finite PRD 50 K \ 10) i5 of order 1 by Theorem 4. Hence K i5 of order 2 50 K = 10. 1). Then the multiplication and addition are determined except for 1 + 1. If 1 + 1 = 0 tben K is the field Z 2 which i5 impossible 5i nce 、ve assumed that K 15 not a field. Hence 1 + 1 = 1 50 K is the Boolean 5emifield. 2) Since K i5 a fini te limited ∞ semifield. K\ {∞) i5 a fi nite PRD so K\ {∞l 15 of order 1 by Theorem 4. Hence K i5 of order 2. Let K = 1 ∞ . 1) . Again the multipli catioD and addition are determined except for 1 + 1. lf 1 + 1 =∞ then 1 1S its own complement which i5 impossible since K is limited. Hence 1+1=1 so K is the almo5t trivial ∞-se m i fi e ld of order 2. COROLLARY. If S i5 a f inile semirù'g ’l1 ith mα ltip [{ca!i ve zero 0 wlu.ch is o-M. C. t h eη S mU5 t be a 5emifield 50 zf Ihe orùr of S is >2 tlw , S nzU5t be a field. PROOF. Sincc S i5 Q.M. C. . (S\ (O) .• ) is a finite cancellati,"c comm Ula ti 、 c semigroup 50 (S\ (O) . . ) is an a belian group and sincc 0 is a multiplicative zcro {)f S. (S . . ) i5 an abelian group with zero. Hence S is a semifield. By Theorcm 5 if the order of S is >2 then S is a fie ld. If K i5 a O'scmifield of order 2 thcn K must bc either Z . or the Boolean "5e mifield. Hence the Boolean semifield i5 the only finite O-semifield whicb i5 not a field. Furthermore. 5ince 1+ 1= 1 in tbe Boolean semificld no enension ,;emifield of the B001ean semifield can be a ficld. Hence every cxtension semifield of the Boolean se mifield must be infinite. Note that in thc Boolean semifield 1 behaves as an additive zero. lt is easil y seen that this cannot happen in any .other O-semifield. If K is an ∞ -se mi f ield of order 2 then K must be either the trivial semifield or the almost trivial semifield 、I>' h 08e multiplication and adcl ition tables are g iven below 334 Sidney S. Mitchell alld Porntip Sz'nutoke 짧몽 띤훌승 홍i훌판 T he almost trivial semifield on { ∞. 1} also shows that it is possible for 0。 semifield to have an additive identity which is not ∞ and again this is not true for any ∞ semifield of order > 2. Also. the almost trivial scmifield on {∞. 1} has + and . that are equal but it is easily seen that no other semifield can ha ve this property. We shall now give yet another application of Theorem 4. DEF1NlT10N. Let D be a PRD. Then the smallest sub-PRD contained in D. i. e. the sub-PRD of D generated by 1. is called the prime PRD 01 D. THEOREM 6. 11 D is an …ifi…te PRD then the prime PRD 01 D is eilher isomorphic 10 Q+ wilh Ihe αsμ al addilion and mulliplicαtìon 0γ is (1). PROOF. Let D’ be the smallest sub-PRD of D. If nEZ 十 de[ine ,,1 = 1 + 1 + .. + 1 (n times). Then {nl} ",,,z.ÇD'. If for all m. nEZ 수 m켜 η implies that 1II1"'nL then one easily 5ees that D'?E Q+ with the usual addition and multiplication. +T Assume that there exist 끼 . nEZ • m > m’ and 11ν l = nl } . Claim that ",' = 1 and n' =2. To prove this. note that if 1/1''''1 or n' 켜 2 then we get that the set {(η1 )(… 1) - 1 1 1 드m ， 1l 드 ，ν → 1 ) is a finite P RD of order > 1 which contradicts Theorem 4. Hence D' = (1). We shall close this section with the remark that all the results on semifields concerning additive cancelIativity remain true f01" PRD’ s. 4. Prime semifields DEFINlTION. Let K be a semifield. Then the prime semilield 01 K is the smallest subsemifield contained in K i. e. it is the subsemifield of K gencrated by 0 and 1. Wc shall now compute all the possible prime semifields for the two types of semifield s. THEOREM 7. 11 K is a O-semilield then Ihe pη me sem,jield 01 K is either tS 0 11l 0η hic 10 Q~ μlith the α sual addilioη and multiplicatio1Z or ζ ω here p is a The Theory o[ Se ηti[ields 335 prime number or the Boolean sem'field. Furthermore. if the prime sα써field of K is isomorþhic /0 Zp then K is a field. PROOF. Let K' be the prime semifield of K. For nEZ+ define η1 = 1 +1十'" + 1 (n times) and D1 =0. Then (η1) .εZ . ‘ 드 K'. If for all m, ηεZ ;; m ￥ n implies that ml ;;:é η 1 then iπt 녕1 s obv、\"10 ~」 plication. Assume that there exist m. nEZ~ such that m (1) Suppose that ， η'=1 and n' =2. Then K' is the Boolean semifield. (2) Assume that m' 톰 1 or n’ ,,0 2. If m' =0 then n'> 1. The case n' = 2 gives K' 르 Z? as a field. If ,,'> 2 then ,,' - 1 > 2 and for all mEZ;; mlE (nll 0르 n드 . ' -1' Also the case nν > 1 giv않 U8 that n'>2 and for all m’11εeZ; ηml 틴타E ，…{ 씨n'l1} 0드 .‘i드i’”“l and for all l ItEZ;; mlE {nli 0 "，.듣 tν 1. Therefore we see that in all these cases n'> 2 and lor all m EZ~' mlE (nl ) 0". 르i → l' Let B = (nllnl ￥ 0 ， nEZ+). Then2 < IBI <∞. Lct C= {(η1)(… 1) ll ml, η1εB ). Again, 2 THEOREM 8. If K is a“ 。o -sem'field then the prime sem'field of K is either isomorþhic to Q! or the trivial semlfield 01 order 2 or the almost trivial sem'field 01 order 2. PROOF. Let K' be the prime semifield of K. Since ∞ +x= ∞ for all xEK, ∞EA where A= (x EK l x+y = ∞ for all yEK). Hence A 7얘. By Corollary 2 of Proposition 2 we have that A = { ∞} or A=K. If A=K then 1+1 = ∞. Thus K’= (∞， 1 ) with the trivial structure. Assume tha t A= { ∞). Let η EZ + and.. de[ine nl=1+1 十 ' + 1(η times) and ∞ 1= ∞ . Hence we havc that {η 1) nEZ.'-;;; κ. If for all 1IZ, η EZ 상，，，0 η l 때 lie s that 찌 l "'n l we then get that K' 즌Q￡ . Assumeι that there exists m, tl εZ;; such that '" Assume that m’,.'1. If m’ =0 and ,,' = 2 then K ’ =1 ∞， 1 ) with the trivia: structure, SUPPo5e that m' =O and ,,'> 2. It is clear that lor aJl I<>n' , k1 = ∞. Since n'> 3, n ’ 2- 2 ，ν + 1 > "'. Hence ((n’•1)) ((n' - 1))1 = (n,2 - 2n ’ + 1 )1 = ∞ which i8 a contradiction since (K\ { ∞ ) ， .) is a group. Therefore this ca5e cannot occur. - 1 Hence m" > l and 50 ,,'> 2. Le t C= l(m1)(nl)- ï m, nEz" Then 2 2. 'Ve claim that there cxists an n εZ + such that η1 has no multiplicative in versc. Suppose that for aJl nεz → η1 ha5 a multiplicative inverse, then C= l( ml )(n 1) - l) ’7I .Hε Z' i5 a finite PRD of order > 1 contradicting Theorem 4. Hence we havc the claim which implies that m’ = 0. a con tradiction siDce m' 二 1. Hencc this case cannot occur also and ,\-'C ha ve the theorem. EXAMPLES. Q~ with the u5ual multiplication and + defined by x 十 y = max Ix, y) is a semifield of zero type and its prime semifield is i50morphic to the Boolean semifield. If we deline x +y = max Ix, y) on Q;;'+ with the u5ual multiplication then we get a semifield 01 infinity type having the alm05t trivial semifield 01 order 2 as its prime semifield. 5. Embedding theorems In this, the last section, we shaJl consider a1l possible cmbedding thco rems b etween semirings. rings, semifields, fields, PRD’ s and integral domains. Before proving the nev.r theorems we shall first review the kno\vn results on embeddings of algebraic systems. 1) A ring R is embeddable in a field if and only if R is O.M. C, (if R has a multiplicative identity then it is an integral domain). Further more, \ve can construct a special field F containing a 5ubring naturally isomorphic to R ca l1 ed the quolienl field of R. F is constructed as follows : !f R = 1 이 let F=Z , . lf R￥ {이 define an equivalence relation "-' on R x (R\ 10)) by ( x, y ) ~(x ’ ， y') if and only if xy’ =x'y. Let F = R X(R\{이 ) /~ and define [(x, y )J + 1(.<', y')l = I( xy' + x'y, x'y') ], I(x, y)J. I(x', y')J = [(xx', yy')J. There is a natural monomo l'phism i: R-• F defined as follows: choo5e bER\ 10). Then if xER define i(x)= [( xb, b) J. The monomorphi5m i i5 independent 01 the choice of bE R\ {이 • (lf R is an integral domain then R has a multiplicative identity 1 50 define i(x) = I(x, l )J for all x ε R). i is called the can이tical injecNon. Thc Tllcory 01 ~e)’uj"ields 337 T he qUGtient field F of R has the tmique extetlsioll property for mO'llomorphisms frolll R i1llo fields i. e. if g : R - • F' is a monomorphism wherc F ' is a field then there i5 a uniq ue monomorphi5m 1 ’ F- • F ’ 5uch that li=g. The uniqu~ extension propeny for monomorphisms gives the fo llowing results a) Every field containing an isomorphic copy of R contain5 an i5omorphic copy of F 50 F is the smallest tïeld containing R in this sensc b) Let R' be a c-M . C. ring, F ’ the quotient field of R', i' the canonical injec tion of R' inw F' and g : R- • R' a monomorphism. Then therc Îs a unique monomorphism 1: F - • F' 5uch thatli=i'g. c) lf F' is a ficid and i ' : R•• .. F' is a monomorphism having the unique extension propeny for monomorphisms from R to fields rhen there is a natural isomorphism ι :F- • F such that 1ti =i' . Note that the previous results imply that a O.M. C. ring R can be extended to an in tegral domain. [n fact, taking the subring of the quotient field generated by RU (1) is the smallest integra l domain containing R. Now if S is a semiring of order > 1 、，vith multiplicati\'c zero, O. which is o-M. C. then the a bo \'e construction can be applied 10 S. 、νe then obtain a semifield K called thc qllotienl semilield 01 S. We also get a callolliml i ，υeclio 11 i: S→一 K \vhich satisfies the unique extension pr때erty for mono:norphisms from S into semifields and al50 properties similar to a), b). c) , abo、 e. \ [oroo‘ e,‘ if S is a semiring of order > 1 which is M. C. then we can apply the a bo \'c construction to S (excepl that hcre we can define the equiyalence relation ...... on S x S) to get a PRD D 、，vhich 、vc shall call thc q μ otìeη t PRD 01 S. Again ‘\.e get a cai!onical injeclioll z": s- • D "\vhich satisfies the un iquc extension property for monomorphisms from S to PRD's and as a resu1t \~ .e get si mi!ar versions of properties a), b). c) abo 、 e . We have thus pro\'ed rhe followii1g theorems: THEOREM 9. Lel S be a semirilzg 10ilh lIIu/tiplicalive zero. Thell S is cmbeddablε ill a semzfield il and ollly il S is 0-111 . C. Fκ rth e r 11l ore. σ S is C-111 . C. Ihen S can be extended 10 a O-M. C. semiriug with multiplicatz' ve ideμμ Iy and Ihe sμ bsemirillg 01 tlze quotienl semil ield geη erα led by SU (1) is 1ft. S1illlllcsl O - jν .C. semirillg with 1nχ ltzþlicalive idenlily contai1ling S COROLLARY. Lel S be a semiring loitlz multiplicalive zero wlziclz is 0-111. C. Then either S is a semirittg with zero Or S is a semiring u;iU‘ ilz/inily. PROOF. By Theorem 9 we can embed S into a semifield. Since e\'ery semifielá 338 Sid l1 ey S. Mt'tchell and Porη tt'p Sinutoke i8 either a O.semifield or an ∞ semifíeld, the mu1tiplicative zero is either a zero {)r an infinity of S. THEORE'I'I 10. Lel S be a semiring. Then S is embeddable in a PRD il a>μl only lJ S is M. C. Furlhermore. il S is M. C. I"en S caη be extellded to an A1. C. sem~nng ω ith mα ltiplicalive idenlily and Ihe sllbsemiring 01 the qα otie 1:μ PRD generaled by SU {1} is Ihe smallesl M. C. semiring with muliip!icative idenlily con!aining S 2) A semiring S is embeddable in a ring if and only if S is A. C. Furthermore we can construct a special ring R containing a subsemiring naturally isomorphic to S called the dillereη ce ring 01 S. R is constructed as fo llows ‘ Define an equivalence relation ~ on SxS by (x, y)~(x' ， y') if and only if x+y' = x' 十 y. Let R = S x S/~ and define [(x , y)] + [(x’ , y')] = [(x 十 x' ， y+y') ]， [(x , y)] ' [(x’ , y’)] = [(xx' +yy', τy' +x'y)]. This is the same as the Grothendieck construction of an abelian group containing a commutat ive cancellative semigrollp except that here ".le have a second binary operation called multiplication. There is a natural monomorphism k : S•• • R defined as follows: choose bES. Then if xES de!inc k(x) = [(x +b, b)]. This monomcr"hism is independent of the choice of bES. The difference ring has the unique extension property for homomorphisms [rom S to rings. i.e. if g : S ←→ R~ is a homomorphism where R~ is a r ing then there is a unique homomorphism f : R••• R' s. t. Ik=g. Moreo、 er ， if g is a mono morphism then f is a monomorphism also. 'The unique extension property for homomorphisms gives properties similar to a), b). c) in 1) (except that in b) we can assume that g is a homomorphism and then 1 will be a homomorphism. If .g is a monomorphism then f will be a monomorphism. The next question that \l,re shall concern ourselves with the follmvi ng under what conditions can ‘ve embed a semiring into an integral domain? Since an integral domain is both A.C. and Q-M.C. both these conditions are necessary. Surprisingly, these two necessary conditioos are oot sufficient. \~T hat is needed is a concept which is stronger than O-M. C. or M. C. DEFINITION , Let S be a semiring. Then S is said to be slγongly multzþlicati l..'ely cancellalive if and only if for a11 xl' x2, YI' Y2ES. xjYj +x,Y,=xjy2+ x2Yl implies -that either x1=x 2 Or Y1 = Y2" 、Ve sha11 abbreviate the words strongly mu1tiplica t ively cancellative by S. M. C. PROPOSITION 4. Lel S be a semiri1lg w서 ch is S.M.C. Thett S is Q-M , C. il S ι as a 11l ι ltiplicatz.ve zero and S is M. C. 1.1 S lzas 1lO mu.!tipUcative Tlze Tlreory 01 Semilields 339 PROOF. Casa 1: Suppose that S has a multiplicative zero. We want to show that S is o-M. C. Assume that %IYl = %IY2 and %1 ",,0. Then %IYl+ 0 'Y2= %IY 2 ~ O 'Y l so by S. M.C. we have that either %1 = 0 or Yl = Y2' However %1 ",,0 SO Y1=Y2' Case 2: Suppose that S has no multiplicative zero. We want to s ho 、、 that S is M. C. Assume that .%'IYl = x 1)'’2 ( 1) Si nce S has no mu1tiplicative zero, there exists a uES such that μ %1 "" % 1' Multiply (1) by " . We get that U%IYl = "x1Y2' Let x2=ux1• Then x 1 잊 X z and X~2 = X2Yl (2) Add (1) and (2). We get that %IYI + %~2 = xlY2+ %2Y2' Since S is S. M. C. eilher 'Xl = x2 or Yl =Y2. However x 1 낯X~ therefore Yl = Y2. PROPOSITION 5. Let S be a semiring which is embeddable in all integral do끼 om or a O-M. C. ring. Th eη S is S. M.C. PROOF. Let "1' %2' YI' Y2ES he such that %IY 1+ x~ 2= xIY2+ x:p’ l ' Since S is embeddable in an integral domain we can consider this equation to be an e quation in an integral domain where we can subtract. Subtracting we get that XIY1-XIY2-%~I + X 2Y2=0 . Hence ( X1- X2)(Y'-Y2)=0. Since %1-"2 and Y'-Y2 are elements in an integral domain, either x1-x2=O or YI - Y2=O. Hence a necessary condition that a semiring be embeddable in an intcgral domain or o-M. C. ring is that it be S. M. C. PROPOSITION 6. There exisls a1t A. C. . M . C. semiring wilich cannot be embedded 씨 an illtegral domain or a o-M. C. rlng. PROOF. By Proposition 5 we need only construct an A. C. , M. C. semiring S which is not S. M. C. Then S cannot he emhedded in any integral domain. Le t + ... 4 S =(Z T)' with the product semiring structure. Tben S is A. C. and M. C. Let x, = (2. 1,1. 1), x2= (1, 2, 1. 1), y,=(1, 1, 2, 1) and Y2=(I. 1, 1, 2) . Then .x 'Yl + X2Y2= %IY2+ X~I=(3 ， 3. 3, 3) and x1' "'2' YI' Y2 are pairwise distinct. Note that if we le~ S =CZ+)4U {(O, 0, 0, 0)) then we get an A.C.. O-M.C. ,semiring which is not embeddabie in an integral domain. We do however ha ve the following theorem. THEORE:>1 11. Lel S be a semiri1lg. If S has a multiþlicalivø idel1H1y 1, the" S 340 Sidlley S. MitcfwU"alld Porntip Sinα toke is embeddable in all inlegral domain ,j and only il S is S. M. C. and A. C. Moreover. zl S has no mκltiþlicalive identity theμ S is embeddable in a O-M. C‘ ring ,j and only ,j S is S.M.C. aη:d A.C. PROOF. By Proposition 5 if S is embeddable in an integral domain then S must be S. M. C. and since an integral domain is a ring S must bc A. C. Conversely suppose that S is S. M. C. and A. C. Since S is A. C. S is embeddable in a ring. Lct R bc thc di[ference ring of S. Claim that R is an integral domain. T o prove this we must show that for all x, yER xy=O implies that x =O Or y=O. Note that thc zerO element 01 R is [(a , a)J where aES. N。、v suppose that x, yER are such that xy= O. Let x = [(x" x2)J and y= [(y" y,)J for some X1, x:';' Yl' Y2ES. Then xIYl + x2Y2=xlY2 + x2Y2 so by S.M.C. X1=X]: or Yl=Y2 i.e~ X 二 o or y =O. '1'0 finish the proof note that [(l +b‘ b)J is a multiplicativfr îdentity in R. ’!,.Ve have alread.y shown that there exists an A. C. semiring '~lhich is not S.M. C. We shall now give an example 01 an S.M.C. semiring 、.vhich Ís not A. C. Hence the h >.,.'o conditions in Theorem 11 are independent of each other. EXA MPLE. Give Z • the usoal multiplication but define addition by ’n + n = min(m, 씨 • Then Z+ is not A. C. but it is S. \1. C. 'iole that in the above prool we showed that the difference ring R 01 an A. C. , S. M. C. semiring S is an integral domain 80 R has the uniquc extension property for homomorphisms from S to rings. Also, the unique extension of a monomorphism from S to a ring is a monomorphism 80 every in tegral domaiIll \vhich contains an isomorphic copy of S must a1so contain an isomorphic copy of R and any ring with this property is isomorphic to R. In this sense we can saγ tbat R is the smallest integral domain containing S. 'Vc shall now study the problem of finding necessary and sufficient conditions. guaranteeing that a PRD can be embedded in a field. Sincc a field is A. C. clearly A. C. is a necessary condition. We shall show that A. C. is not sufficient, anotber property defined below, is needcd to quarantee sufIiciency. DEFIKITlON. Let D be a PRD. Then D is said to be þrecise if and only if for all μED 않v + 냄r = 1 iI1lPIles that α = 1 Or v = l. The Theory of Semifie/ds 341 PROPOSlTIOK 7. Lel D be a PRD which can be embedded in a field. Thetl D is þrecise. PROOF. Suppose that we ha 、 e tWQ elements ι .v ED such that ~+-，--!!-- 1+ μ V I 1 +ιu = 1. Then ,‘ +v=l +u". Since D is embeddable in a fie ld we can consider this equation to be an cquation in a field 、，vherc we can subtract. Subtracting gives us that 1 “ ν +uo=O therefore (1-,,)(I -v)=O. Since these are elements in a field eithcr u=l or v= 1. 50 we sec that a necess퍼 ry condition that a PRD be embeddable in a field is that it be precise. PROPOSITIO'l 8. There exisls an A. C. P RD 10Mch camlOl be embedded in a field. PROOF. By Proposition 7 we need only construct an A. C. PRD which is not precisc. Let D=Q+ X Q ι and gi,-e D the product PRD structure. Then D is an A.C. Pl Howe‘ CI “ e do have the foIIowing theorem. THEOR EM 12. Lel D be a PRD. T h.,. D is embeddable in a field if alld ollly 'J D is A.C. and precise. Furtllermore. 'J D is a1l A. C. alld preäse PRD 111m its dzfference ring R is a η {ntegrα 1 domai11. W c shall call the qtlolielll f ield F of R the differellce field of D PROOi'. Suppose that D is embeddable in a field. Since a ficld is A. C. . D is A. C. By Proposition 7. D must also be precise. Conversely. suppose that D is A. C. and precise. Sincc D is an A. C. semiring we can embed it in its difference ri ng R. Claim that R is an integral domain. This cJaim s ho、v s that D is embeddable in a field since an integral domain is embeddable in a field. so suppose that α ， βER are such that aß=O. We must show that cither α =0 Qr ß=O. Choosc (0. b)Eα and (c. d)Eß. Then α . b. c. dED. Since α/9=0. ac +bd = -1 -1 •1. -1 ad+bc. MuItiplybothside50fthisequationby a- COROLLARY 1. Lel D be a1l A.C.. precise PRD al!d F ils diffeTe1lce field. T Jz.en theTe is a natural 1nonomorþltism z' : D-• F salisfyi1lg 111. μ 1lÎque extension 342 Sid lley S. Mitclzell and P or ll t ψ Silt :J. toke þroþerly for mOl!olllOrþ"is/lts f r01ll D 10 fields i. e. give“ a m otlo까orþ h ism g : D F' w/z ere F' is a field lhere exisls a μ ll iq u e mOllomorþhism f: F - F ’ such tllat I"e f ollo“'ing diagram com mutes f F-• F' 2\ / g D ROOF. There is the canonical monomorphism ;1 from D to its ι iffere ; : cc rÎng R which is an integral domain and a canooical monomorphism μ from R to it5 quoticnt field F which i5 the diffcrence field of D. Let i= ’ 2~. 1' Then i is a monomorphism from D to F. We mU5t sho w that (F . i) has thc un iquc extension pro pe rty for monomorphisms from D to fields. 50 lct g : D- .F' be a mono. morphism where F' is a field. Since F' is a fi eld . F’ is a ri ng 50 there is a uniq ue monomorphism gl : R•• F ’ such that g,i,=g. Since F' is a fi eld a nd R is an in tegral domain there is a uniq ue monomorphism g2: F ~F ' such that g션 2 = gr Now gzI =g션: 2; 1 =g/l =g 50 there exists a monomorphism g2 : F - • F such that g.j=g. Lastly we must pro\'e thc uniqueness of g2' Let " ‘ F - • F be a monomorphism such tha t Iti =gi . \νe: must show that h=g2' Now"i =g.j 서 t‘ ， =g션2’ ， = g ， i ， =g. By the uniqueness of the extension of homomorphisms for difference rings we ‘.et that "μ 2=g ，. Now g,=g.j2 so hi 2 = g장 2' By the uniqueness of the exlension of monomorphisms fo r quotien t fields w e get that h=g2' COROLLARY 2. Lel D be an A. C. . ρrecise PRD, F its differellce f ield a71d i Ihe ’notwmor phis11l 170m D 10 F give’‘ ;11 Corollary 1 ( lOhich we shall call I"e catlonical i1ljection Iro’11 now Otl). The1t 1) any f ield wlzich c01ztaitlS a1l iso 11lorphic copy 01 D must also conlaitt al1 ’somorþhic coþy of F (so F is Ihe smallesl fie/d cOlltainitzg D itl t ι is se"se). 2) if D’ is a1l0lher A.C" þrecise PRD willl differellce field F' a1ld callonical …~jection ì' mzd f : D- • D' is a monomorPhism then there u a umq“c ’nono 11lorþhlsm g ‘ F ~ F' suc" thallhefollowing diagranz conznzμ les: F조. F ' i 1 1i' D- • D ’ f 3) if F1 is afield a1zd ;1 : D- F1 is a monomorþJz ism flav ittg the unique extettS io" þroþerly f or 1It01l0morþhis11ls f rom D 10 fields. Ih" , Iltere is a 1101μ ral isol1lorþltism The Theory 01 Semilields 343 h: F•• F , sα ch thal the followittg diagram cmnn…tes F-• F, z\ /i ，~ D PROOF, 1) Let H be a field containing an isomorphic copy of D. Let g: D -• H be a mo nomorphism from D onto its isomorphic copy in H , Then there exists a monomorphism f : F•• H extending g so 1mσ) is an isomorphic copy of F in H. 2) i'f: D-→ F~ is a monomorphisrn 80 there exists a unique monomorphism g :F- • F' such that g i = νj. 3) i, ‘ D- • F 1 is a monomorphism 50 there exists a unique monomorphism h :F•• .F, such that hi= i,. i ,D-• F is a monomorphism 80 there exists a unique monomorphism k: F ，-~ F such that ki， 二 i. No'iV khi=ki ， 二 i=1 β. By the uniqueness of extensions. kh = lp SimiIarly hk=1 κ 80 F is naturaIIy isomorphic to F 1 We have already shown that there is an A. C. PRD which is not precise. We shaJl now give an example of a precise PRD which is not A. C. Hence the two conditions in Theorem 12 are independent 01 each other EX AMPLE. Give Q+ the usual multiplication but dcfine addition by r , s=min {r. s}. Then Q+ is not A. C. but it is precise PROPOSITIO'-í 9. Let S be an M. C. semiring 01 order > i. Tilen thc qiω tieηt P RD D 01 S is precise ,j and 01lly il S is S. ‘M.C. PR OOF. Supposc that S is S. M. C. Let α ， βED ha ve the property tha t ←뜨→ 十 ←4r= l. Then l ÷ α ，8=α 녕. Let α = 츄 and /3 = 관、vhcrc x ， χ ! ， )', 1 ， α ß I l , aß Thcn 1+ 80 X' + x'y. Sincc S is y 갚 S. xy.~ζ= 스x + +=y xfxy그 fι l' 十 xy = xy' S.M.C. either x = x' or y =y' i.e. either α = 1 or ß= 1. Converscly, suppose that D is precise. Let xl' xz' Yl' Y2 E S be such Lhat X 1Y l 十 X2Y2=XIY2+ X2Y !" Considcr this to be an equation in D and divide by x j Y1' '^le X2 Y2 Y2 x2 .... , __ r. ,_ _ _ ~_:~ _ _: ... 1__ __ x2 1 _ __ Y~ get that 1 + ← --~- = --~-+ ←←. Since D is precise ei ther ~= 1 or ~ =l Xl Y1 Y1 x 1 λ y ↓ i. e. either x1 =xz or Yl =Y2' 344 Sidney S. 111itchell a1Z d Por씨 iþ S:o Jl utoke The results that we have just obtained for PRD’s can be simiJa rly obtained for semifields. DEFIJ\ITION. Let S be a semiring. Thcn S is said to be þrecise if and only if for a11 11 . vES 1 +,'"=.‘ +v implies that ι= 1 Or v= 1. PROPOSITION 10. Lel S be a semifield Ihat is embeddable ill a field. Then S zs prectse. PROOF. Same proof as Proposition 7. PROPOSITlON 11. There exisls an A. C. sem;f;eld IOhκ" call1101 be embedded i7! a f;‘eld. PROOF. Take Q~XQ+U((O ， 0)). THEOREM 13. Let K be a semifield. Then K ;s embeddable ;n a fielà ;f and 。nly '1 K is A. C. alld precise. Furlher…ore. if K is an A. C. and þrecise semifield theη ils d.lfereπce ring R is att integral domain. We sl:all call Ihe quolielll field F of R tlze d,lfennce field of K . There is a natural nW/lolIZ orþhism i: K-• F (whicl.‘ we sllall call the caη olu"cal i η:j ec/ z’ on) salisfying Ihe μ nique extensio’‘ proþerty lor mOIl()morpltis1lls 17011‘ K 10 fields. PROOF. The proof is similar to the proof of Theorem 12. As a result of Theorem 13 we see that 1) any field containing an isomorphic copy of an A. C. , precisc semifield K must also contain an isomorphic copy of its difference field F so in this sense we can say that F is the smalIest field containing K. 2) if K , K ’ are A. C. precise semifields with difference fieJds F , F' and canonical injections i,;' respectively and if f: K-K' is a monomor phism thcn there is a unique monomorphism g : F-• F' SUC!l rhat i'f=gi. 3) 11 F, is a fieJd and ;1 ‘ K-• F1 is a monomorphism satisfying the unique extension property for monomorphisms from K to fields then there is a natural isomor phism " : F-• FI such that 11;=;1' Thc two conc띠l(띠뼈dl ‘w띠it야th thc usual mu비j끼lt“iψp미lication but define addition by r+s=max‘ (rκ’ s야). Then K is precise but not A. C. W녕e ha、v’ e 외a Iready given an example of an A. C. semifield which is not precise. PROPOSITIO:-I 12. Let S be a sentiring ψ itiz ’1lZtlliþlicative 2ero wlzich ;s O-M.C. and of OI'der >1. Theη IlIe quotietll sem;f;eld K of S is precise if and ollly if S Tlw Theory of Semifj‘elds 345 is S.M.C. PROOF. Sim ilar to the proof of Proposition 9. Com bining all these embedd ing theorems we get the following result. PROPOSITIO)! 13. Let S be a se:lliritlg. Th eη S is embeddable jll a field 1I and ollly jf S is A. C. at찌， S. M. C. PROOF. By Theo rem 11. S is embαlda b le in an integral domain if and only if S is A. C. and S. M. C. Since a field is an integral domain both conditions are necessary. Since an integral domain is embeddable in a fielù both conditions are sufficicnt. As a result of Proposition 13 we can embed an A. C . • S. M. C. semiring S in a field by first constructing the difference ring R of S (which we showed is an integral domain) and then cmbedding R into its quotient fielù . The next question that we wish to consider is the following : can we embed an A. C. , S. M. C. semiring into a field by first embedding it into a semifield and then embedding the semifield into a field? To answer this question, notc that since S is S. !vl. C. S is 0• M. C, (or M. C. if S has no multiplicativc zero ) so we can surely embeù S illto its quotient semifielù K. By Proposition 12. K is precise since S is S. \1. C. However, ‘\'e ne밍d mOre than this to be ablc to embed K in a field , we must have that K is also A. C. This is obtained from the foll。、，ving pl'oposjtion. PROPOSITIOl\ 14. Lel S lJe a semi,;,‘g whiclz is A. C. alld O.M. C. (or iν . C . if S has 110 t.시uUiþiicat ω e 2ero). Th eη the qltotiellt semifield ( PRD) of S is A. C. PROOF. Let J( denote thc quotient semifield of S. Assul1lc that a , ß, rεK are such that a+r=,6+r, Wc must show that a= ß. Choosc (x, x’ ) E a . (y ，)" ) εβ and (2, 2’)Er. Then x, y, 2εS ， x', y’, 2’ES\{O} and we sball 、、，r rite α ， β. 1' m the form α = 수 β=4- ， r =4. Now 4+4=4 十」ξ m 뜨류감= Y z x' z' Y' 2 λ ι and therefore 간:±1:εy z 표뿌ξ 」펀:ε Mu비J!빼t tion by x'y'. We then get that X2γ +x ’ 2y ’ =x'yz' ..L. xγ2. Since S is A. C. X2γ= X'Y2'. Since S is O-M.C. and 2' 7" 0 we get that xy'= x'y s。 추=웅 ， i. e. a=ß. As a result of Proposition 14 the quotient semifield (PRD) o[ an A. C .. S. M. C. semiring will be both precise and A. C. and hence by Theorem 13 is embeddable in a field. So we can embed an A. C.. S. M. C. semiring S into a 346 Sidlley S. μ itche l/ aJ1 d POTuliþ Sin χ t ok e field ,y first embedd ing S into its quotient semifield K and then embedding K into its diffcrence field. As a result. if S is an A. C .. S. :V!. C. semiring then we ha 、 e two 、 \'ay s of embedding S into a field. First wc can take the path prescribd by Proposition 13 giving us a field which we shaìl denote by F , and second ly we can take the path prescribed by the preceding paragraph gi\'ing us a lïclJ whi ch we shall denote by F 2• 18 F 1 isomorphic to F 2? ,!'he answer to this queslion is yes as the fo Jlowing theorem shows. THEORDl 14. There are 1IIonolllorþhisllls jt : S- • F , a",} j 2 : S-• F 2 s-u cll lhal (F l' j ,) and ( F 2• j 2) bolJz ha ν e lIze u niqκ e exleη s l on propcηy f or JJW1lOl'Uorþhisms from S 10 fields we shall call j l' j2 the canonical injections. As a res!tlt. 1) Fj is 1zalt‘rally isomorphic 10 F 'J' in fact , there z"s a llalural iso1Jlorþhism h : F 1--F'? S Iκ， h tlzat hjt =j 2' 2) Any field which contaÎ1ts an iso7norþllic copy 0/ S 1Jlust also co!ttailz an isomorþhic cOþy of F , and F 2 so I"e field oblained ;'1 bolh ways is Ihe smallest field contai’”’zg S i’‘ this se"se. 3) Any pα ir (F , j ) sα tisfying the U1zique exlellsi011 propeγty f or mo1tomor Þι iS 11l S Irom S /0 a field is llalura/ly isomorpllic 10 (Fl' j j) alld (F 2• j2) in Ihe same way tlwl ( F l' j) alld (F 'l' j ) are ,wlurally isomorpizic 4) If S is all r1. C.. S. lVI. C. semirillg α7띠 g ‘ S → Sι 1$ a 1uono’1l0r pllisnt tlzen tllere is a κ n ique m01l0mor phism f: F 1 ‘F ’ sucll tha! i'f = gi where F ’ is tlze f ield oblailled from S’ i1z lhe sωIJl e way flzal Fj orF2 ’s ob/ailled I rom S a1ld ‘ is lhe ’‘atural injec/ioll. PROOF. If S= {O) thcn the proof is trivial, the field wc gcl is Z ,. so \\'0 nJOj assume that order 0 1" S is > 1. Let R be the differencc ring of S and 서 S-.R the canonical injection. Then F , is the quotient field of R. Let i2 ‘ R - 'F, bc che canonical injection and define j l =;_서. Thc prooî that ( F l' μ ) has the unique extension property fo r monomorphisms from S to a fjelò is sim ilar to lhe proof given in Co rol lary 1 of Theorem 12 Let K be the quotient " mificld of S and kj : S-• K the canonical injection. Thc F 2 is thc differe!lce field of K. Lct k2 : K - • F 2 be thc canonical injection and dcfine j 2=k2k j• The proof that (F2• j 2) has the uniquc eχ tcn sio n property for monomorphisms from S to a ficld is similar to the proof gi vcn in Corollary 1 of T l1eorem 12. Thc remainder of the proof is the samc as the proofs of Coroliaries 1 and 2 of T hcorem 12. The Tlzeory 0/ Se lJllfields 3.17 We shaIl caIl the field obtained in Theorcm 14 the 111;'씨 1/( 111 Jield oJ S. \V hul ‘,ve have done is generalize the construction of Q from Z:. + z。 /' \、 Q;;3\/ Z Another example of the construction of the minimum fi elù is the following. Let Z :IXl = σEZ6' IXll all thc coefficients of J < deg J are nonzero). Wc call Z : IX) the res성trtκCμted po띠ly’씨， 1 be the quotient PRD of the M. C. scmiring Z: IX). We shall cal1 Q: CX ) thc restricted PRD of rational maps with coefficients in 잃. Thcn the differcnce ring of Z : IXl is Z [Xl and the minimum field of Z : [X] is Q ( X ). Z ; [X] /' \、、‘ Q:CX ) ZIXl \ QCX )/' Notc that in going from Q: CX) to QCX ) thcre arc two dis linc t s 때 s. 'r‘ he difference ring of 와 CX) is not QCX ) sincc τ융 does not bclong to it. As a last example let 2Z= (21l lnEZ) and 2Z; = 1 2씨 tl E Z인 . Tbcn we get the foIlowing diagram 2zr /' \、 Q6' 2Z 、'- Q /' Chulalongkorn University, Bangkok 10500 , Thailand. J.1S Sidlle)' S. MitcJzcll alld Pornliþ Sinutoke !lEFERENCES [1) Dale, L. and Picts, J. D. , E“clitieoll Gnd Gaussiall semirings, Kyungpook Mathema tical Journal 18(1 978) , 17-22. [2] Stone. H. E. . il1atrix reþresentatiou 01 sz"mþle halfrings, Transactions of thc Amer. ~In[h . Soc. 233(1977), 339-353. [3) _←---←~ldιals iJ‘ Izalfrings, Proc. Amer. ~'la th . Soc. 33(1 972), 8-14