Simple Commutative Semirings

Journal of Algebra 236, 277᎐306 Ž2001. doi:10.1006rjabr.2000.8483, available online at http:rrwww.idealibrary.com on

Simple Commutative Semirings R. El Bashir, J. Hurt, A. Jancarık, ˇ ˇ´ and T. Kepka1 Faculty of Mathematics and Physics, Charles Uni¨ ersity, Sokolo¨ ska ´ 83, 186 75 Prague 8, Czech Republic E-mail: [email protected]; [email protected]; [email protected]; [email protected] Communicated by Kent R. Fuller Received February 1, 2000 Key Words: commutative semiring; cancellative semiring; parasemifield; semifield.

0. INTRODUCTION The notion of a semiring Ži.e., a universal algebra with two associative binary operations, where one of them distributes over the other. was introduced by Vandiver w33x in 1934. Needless to say, semirings found their full place in mathematics long before that year Že.g., the semirings of positive elements in ordered rings. and even more so after Že.g., various applications in theoretical computer science and algorithm theory.. However, the reader is referred to w11, 12, 16, and 17x for background, basic, and more advanced properties of, and comments, historical remarks, and further references on semirings. Congruence-simple algebras Ži.e., those possessing just two congruence relations. serve a basic construction material for any algebraic structure. In spite of the fact and in contrast to the enormous and opulent supply of

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While working in this paper, the first and fourth authors were supported by the grant agency of Charles University, Grant 3051-10r716, and the second author was supported by the grant agency of the Czech Republic, Grant 201r97r1176. The first author was also supported by the grant agency of the Czech Republic, POSTDOC Grant 噛201r98rP247. The authors were also supported by the institutional grant CEZ: J13r98: 113 200 007. 277 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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information on worldwide popular simple groups and rings, not much is known on congruence-simple semigroups and almost nothing on such semirings. We mention only that the study of congruence-simple commutative semirings with unit was initiated in w26x Žsee also w12x. and that some results on congruence-simple semifields was achieved in w21x. The aim of the present paper is to classify the congruence-simple commutative semirings and to relate them with better known concepts Žfor instance, ordered rings with order units.. To that purpose, the paper is divided into 14 parts and the promised classification is summarized in the 10th section. Additionally, for a better understanding, some basic information on ideal-simple commutative semirings is included.

1. PRELIMINARIES A commutative semiring is a non-empty set equipped with two associative and commutative binary operations Žusually denoted as addition and multiplication. such that the multiplication is distributive over the addition. ŽNote that the existence of any neutral andror absorbing element is not assumed a priori.. In the following we shall be handling commutative semirings only, and hence the word semiring will always mean a commutati¨ e semiring. Let S be a semiring. A Žbinary. relation r : S = S is a congruence of S if r is an equivalence and Ž a q c, b q c . g r, Ž ac, bc . g r for all Ž a, b . g r and c g S. A non-empty subset I of S is an ideal or bi-ideal of S if SI : I and I q I : I or S q I : I, respectively. The semiring S will be called congruence-simple Žor cg-simple, for short. if S is non-trivial and id S , S = S are the only congruences of S; and ideal-simple Žid-simple for short., or bi-ideal-simple if S is non-trivial and I s S whenever I is an ideal or bi-ideal of S such that I contains at least two elements, respectively. If I is a bi-ideal of S, then Ž I = I . j id S is a congruence of S and consequently S is bi-ideal-simple provided that S is cg-simple or id-simple. Note that a Žnon-trivial commutative. ring is congruencerideal-simple Žas a semiring or a ring. if and only if it is a field or a zero multiplication ring of finite prime order. Note also that every two-element semiring is both congruence- and ideal-simple and that there exist Žup to isomorphism. just eight two-element semirings Žsee the next section.. A semiring S is said to be additi¨ ely Žresp., multiplicati¨ ely . idempotentr cancellati¨ e if so is the additive Žresp., multiplicative. semigroup SŽq. Žresp., SŽ⭈... These semirings will also be called airac-semirings Žor mirmc-semirings..

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2. TWO-ELEMENT COMMUTATIVE SEMIRINGS Z1

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3. CONGRUENCE-SIMPLE COMMUTATIVE SEMIRINGSᎏBASIC CLASSIFICATION 3.1. THEOREM. Let S be a congruence-simple semiring. Then just one of the following three cases takes place: Ž1. Ž2. Ž3.

S is a two-element semiring isomorphic either to Z1 or Z2 , S is additi¨ ely idempotent. S is additi¨ ely cancellati¨ e.

Proof. First, define a relation r on S by Ž x, y . g r if and only if 2 x s 2 y. Then r is a congruence of the semiring S and we have either r s id S or r s S = S. Let r s id S . The mapping x ¬ 2 x is an injective transformation of S and we shall define a relation s on S by Ž x, y . g s if and only if there exist a non-negative integer i and elements u, ¨ g S j  04 such that 2 i x s y q u and 2 i y s x q ¨ . Again, s is a congruence of S and Ž x, 2 x . g s for every s g S. Consequently, if s s id S , then SŽq. is idempotent and hence we shall assume that s s S = S and a q b s a q c for some a, b, c g S. There exist i G 0 and w g S j  04 such that 2 i b s a q w and we have b q 2 i b s b q a q w s c q a q w s c q 2 i b. Then 2 b s b q c for i s 0 and, if i G 1, then 2Ž b q 2 iy1 b . s 2Ž c q 2 iy1 b ., b q 2 iyb s c q 2 iy1 b, and 2 b s b q c by induction. Quite similarly, b q c s 2 c, and so 2 b s 2 c and, finally, b s c. We have proved that SŽq. is cancellative.

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Now let r s S = S. Then there is an o g S such that 2 x s o for every x g S. Similarly as above, define a congruence t of S by Ž x, y . g t if and only if 3 x s 3 y. If t s id S and a q b s a q c, then x ¬ 3 x is injective and the equalities 3b s b q o s b q a q a s c q a q a s c q o s 3c imply b s c Žhence S Žq. is cancellative .. Therefore, assuming that t s S = S, i.e., x q o s o for every x g S. We have to consider the following four cases: Let S q S s S and SS / o. Then there are a, b, c, d g S such that a q b s c and cd / o. Since xo s x Ž o q o . s xo q xo s o for every x g S, we know that all the elements a, b, c, d are different from o. Put I s  x; x s cu q ¨ , u g S, ¨ g S j  044 . Then I is a bi-ideal of S and o, cd g I. Thus I s S and, in particular, a s cu1 q ¨ 1 and b s cu 2 q ¨ 2 for suitable u1 , u 2 g S and ¨ 1 , ¨ 2 g S j  04 . But then a s Ž a q b . u1 q ¨ 1 s au1 q ¨ 1 q ¨ 2 u1 q cu 2 u1 , b s Ž a q b . u 2 q ¨ 2 s bu 2 q ¨ 2 q ¨ 1 u 2 q cu1 u 2 , and o / c s a q b s au1 q ¨ 1 q ¨ 2 u1 q bu 2 q ¨ 2 q ¨ 1 u 2 q 2 cu1 u 2 s o, since 2 cu1 u 2 s o, a contradiction. Let S q S / o s SS. We have a q b s c / o for some a, b, c g S and the relation p defined on S by Ž x, y . g p if and only if a q x s a q y is a congruence of S. Of course, Ž a, o . g p, p / id S , p s S = S, Ž b, o . g p, and o / c s a q b s a q o s o, a contradiction. Let S q S s o / SS. Put J s  x g S; xS s o4 . Then J / S and, since J is a bi-ideal of S, we must have J s  o4 . Put T s S _ o4 , and, for every a g T, define a relation h a on S by Ž x, y . g h a if and only if ax s ay. Then h a is a congruence of S, Ž b, o . f h a , for at least one b g T and consequently h a s id S . Thus TT : T and the relation qT s ŽT = T . j id S is a congruence of S. Since qT / S = S, we have qT s id S , cardŽT . s 1, and S ( Z2 . Let S q S s o s SS. Then every equivalence defined on S is a congruence of S, and therefore < S < s 2 and S ( Z1. 3.2. EXAMPLE Žcf. w26, 4.1x.. Let G be an abelian group Ždenoted multiplicatively., o f G, and V Ž G . s G j  o4 . Put x q y s o, x q x s x, and xo s o s ox for all x, y g V Ž G ., x / y. The V Ž G . becomes a Žcommutative. ai-semiring Žpossessing a unit. and o is a smallest Žalias absorbing. element of the semilattice V Ž G .Žq.. Moreover, it is easy to see that V Ž G . is both congruence- and ideal-simple. Note also that V Ž G . is a finitely generated semiring if and only if the group G is finitely generated and that V Ž G 1 . ( V Ž G 2 . if and only if G 1 ( G 2 . 3.3. THEOREM Žcf. w26, 4.7x.. Let S be a congruence-simple additi¨ ely idempotent semiring. Then just one of the following three cases takes place: Ž ␣ . S is isomorphic to one of the two-element semirings Z3 , Z4 , Z5 . Ž ␤ . The additi¨ e semilattice SŽq. possesses a smallest element o Ž i.e., x q o s o for e¨ ery x g S ., So s o, G s S _ o4 is an Ž abelian. subgroup of

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the multiplicati¨ e semigroup SŽ⭈., and the semiring S is Ž isomorphic to . the semiring V Ž G . Ž see 3.2.. Ž␥ . The semiring S is multiplicati¨ ely cancellati¨ e. Proof. Put A s  x g S; cardŽ Sx . s 14 . If a g S _ A and if r a is defined by Ž x, y . g r a if and only if ax s ay, then r a is a congruence of S, r a / S = S, r a s id S , and the map x ¬ ax is an injective transformation of S. Now, if A s ⭋, then Ž␥ . is true, hence assume that A / ⭋. One sees easily that there exists an element w g S such that SA s w s Sw and the set S q w is a bi-ideal of S. If S q w s S, then w s 0 is a neutral element of SŽq., ŽT = T . j id S is a congruence of S, where T s S _ 04 , and we have < T < s 1, < S < s 2, an either S ( Z4 or S ( Z5 . Hence, assume that S q w s w and w s o is a smallest element of S Žq.. Now, define a relation s on S by Ž x, y . g s if and only if  z1 g S; ax q z1 s o4 s  z 2 g S; ay q z 2 s o4 for every a g S. Then s is a congruence of S and we consider first the case s s S = S. We have Ž x, o . g s for every x g S and it follows that u¨ q z s o for all u, ¨ , z g S. In particular, u¨ s u¨ q u¨ s o, i.e., SS s o and A s S. If a g S _ o4 , then S q a is a bi-ideal of S, a, o g S q a, hence S q a s S, a is a neutral element of SŽq., < S _ o4< s 1 and S ( Z3 . Next, assume that s s id S . Then A / S and, if z g S _ A and ¨ g A, then z¨ s o s zo implies ¨ s o. This means that A s  o4 and that G s S _ o4 is a subsemigroup of S Ž⭈.. Clearly, G is a cancellative semigroup. Now, let a, b g S, a / b. We are going to show that a q b s o. Proceeding by contradiction, assume that a q b s c / o. Since a / b, we may also assume that c / a. Then Ž a, c . f s s id S and there exist d, e g S such that dc q e s o / da q e; we have d / o and dc / o. Further, denote by B the set of x g S such that x q ¨ dc s x for some ¨ g S j  14 . Then o, dc g B, B is a bi-ideal of S, B s S, da g B, and, finally, da q wdc s da for some w g S j  14 Žin fact, da q dc s dc / da, and so w g S .. Quite similarly, da q uŽ da q e . s da for some u g S j  14 . Now we have da q uda s da q u Ž da q e . q uda s da q uda q ue s da q u Ž da q e . s da, da q wudc s da s uda q uwdc s da q u Ž da q wdc . s da q uda s da, wdc q wda s wda q wdb q wda s wda q wdb s wdc, wda s wda q wuda q wue, da q wue s du q wdc q wue s wue q da q wdc q wda s wue q da q wdc q wda q wuda q wue s da q wdc q w Ž da q uda q ue . s da q wdc q wda s da q wdc s da.

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Finally, o / da s da q da s da q wudc q da q wue s da q wu Ž dc q e . s da q wuo s da q o s o, a contradiction. We have proved that a q b s o for all a, b g S, a / b. From this, it follows easily that Sa is a bi-ideal of S for every a g S. If a g G, then < Sa < G 2, and therefore Sa s S. Now, it is easy to conclude that G Ž⭈. is a group and S ( V Ž G .. 3.4. THEOREM. Let S be a congruence-simple additi¨ ely cancellati¨ e semiring. Then just one of the following three cases takes place: Ž ␣ . S is a zero-multiplication ring of finite prime order; Ž ␤ . S is a field; Ž␥ . The semiring S is multiplicati¨ ely cancellati¨ e and the additi¨ e semigroup SŽq. possesses no neutral element. Proof. First, assume that there exists a neutral element 0 g S for the additive semigroup SŽq., denote by M the set of invertible elements of SŽq., and define a relation r on S by Ž x, y . g r if and only if y s x q u for some u g M. Since S Žq. is cancellative, we have S0 s 0, SM : M, and it follows easily that r is a congruence of S. If r s S = S, then M s S, S is a ring, S is ideal-simple, and it is clear that either Ž ␣ . or Ž ␤ . is true. Now, let r s id S . Then M s 0 and we put q s ŽŽ S _ 04. = Ž S _ 04.. j id S . For every a g S, the relation r a defined by Ž x, y . g r a if and only if ax s ay is a congruence of S, Sa s 0 if r a s S = S, and Ž S _ 04. a / 0 if r a s id S . Using these observations, we deduce easily that q is a congruence of S. But then q s id S , < S < s 2, and either S ( Z 7 or S ( Z8 . Now, assume that 0 f S and put A s  x g S; < Sx < s 14 . Proceeding in a manner similar to the proof of 3.3, we show that A / ⭋ and that SŽ⭈. is cancellative.

4. ADDITIVELYr MULTIPLICATIVELY CANCELLATIVE COMMUTATIVE SEMIRINGSᎏBASIC OBSERVATIONS The results of this section are of auxiliary character; all are fairly basic and some of them may be considered more or less folklore. We shall not attribute them to any particular source. By a parasemifield we will mean a non-trivial Žcommutative. semiring S such that the multiplicative semigroup SŽ⭈. is an Žabelian . group.

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4.1. PROPOSITION ŽThe Parasemifield of Fractions.. Let S be a non-tri¨ ial mc-semiring. The there exists a parasemifield P Žs P Ž S .. such that the following conditions are satisfied: Ži. Žii. Žiii. Živ. Žv.

S is a subsemiring of P and P s  aby1 ; a, b g S4 . P is additi¨ ely cancellati¨ e if and only if S is. P is additi¨ ely idempotent if and only if S is. If S is congruence-simple, then P is. P is ideal-simple Ž in fact, P does not possess any proper ideal ..

Proof. The proof is standard and easy. 4.2. PROPOSITION ŽThe Ring of Differences.. Let S be a non-tri¨ ial ac-semiring. Then there exists a Ž commutati¨ e . ring R Žs RŽ S .. such that the following conditions are satisfied: Ži. S is a subsemiring of R and R s  a y b; a, b g S 4 . Žii. R is a ring without zero di¨ isors if and only if ac q bd / ad q bc for all a, b, c, d g S; a / b; c / d. Žiii. R possesses a unit if and only if there exist a, b g S such that ax q by q y s ay q bx q x for all x, y g S. Živ. If S possesses a unit, then R does also. Žv. R is a field if and only if for all a, b, c, d g S; a / b; there exist x, y g S such that ax q by q c s ay q bx q d. Proof. The proof is standard and easy. A non-trivial additively cancellative semiring satisfying the equivalent conditions of 4.2Žv. will be called conical. 4.3. PROPOSITION.

Let S be a non-tri¨ ial ac-semiring.

Ži. If 0 f S and S is conical, then S is multiplicati¨ ely cancellati¨ e and P Ž S . Ž see 4.1. is also conical. Žii. If S is a parasemifield, then S is conical if and only if for e¨ ery a g S, a / 1, there exist x, y g S such that a q x s 1 q ax q y. Žiii. If S is congruence-simple then either S is conical or S is a zero-multiplication ring of prime order. Proof. The proof of both Ži. and Žii. is easy. To prove Žiii., let I be an ideal of the ring R s RŽ S . Žsee 4.2. and define a relation r on S by Ž x, y . g r if and only if x y y g I. Then r is a congruence of S, I s 0 if r s id S , and I s R if r s S = S. Thus R is ideal simple and the rest is clear.

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4.4. LEMMA. Let S be a non-tri¨ ial ac-semiring such that 0 f S, and let w g R s RŽ S . be such that w 2 g S. Put S1 s  a q bw; a, b g S j  04 , a q b / 04 and S2 s  a y bw; a, b g S j  04 , a q b / 04 . Then: Ži. Žii. Žiii.

Both S1 and S2 are subsemirings of R and S : S1 l S2 . If 0 g S1 l S2 , then cw s 0 for at least one c g S. If S is an mc-semiring, then either 0 f S1 or 0 f S2 .

Proof. If a1 q b1w s 0 s a2 y b 2 w for some a1 , a2 , b1 , b 2 g S then yb1 b 2 w 2 g S and b1 b 2 w 2 g S, a contradiction with 0 f S. Furthermore, if w s e y f, e, f g S, and cw s 0, c g S, then e / f Žsince w / 0. and ce s cf. In the remaining part of this section, let S be a non-trivial amc-semiring. Then SŽq. contains no neutral element Žsince otherwise S0 s 0. and the relation FS defined on S by x FS y if and only if y s x q z for some z g S j  04 is an ordering; this ordering is compatible with respect to both the addition and the multiplication of S. Moreover, FS can be extended to an ordering Ždenote it also by FS . of the difference ring R s RŽ S . Žsee 4.2.; we have x FS y in R if and only if y y x g S j  04 . The following lemma is obvious: 4.5. LEMMA. Ži. S is upward-cofinal in RŽFS .. Žii. The following conditions are equi¨ alent: Žii1. FS is linear on R. Žii2. FS is linear on S. Žii3. For e¨ ery u g R, u / 0, either u g S or yu g S. Žii4. S is semisubtracti¨ e. ŽThat is, for all a, b g S, a / b, there exists x g S such that either a q x s b or b q x s a.. 4.6. LEMMA. Let S be an ac-parasemifield. If a, b g S are such that b 2 FS a2 , then b FS a. Proof. We have a2 s b 2 q z for some z g S j  04 , and then aŽ a q b . s a2 q ab s b 2 q z q ab s bŽ a q b . q z s Ž b q z Ž a q b .y1 .Ž a q b .. Consequently, a s b q z Ž a q b .y1 , z Ž a q b .y1 g S j  04 , and b FS a. 4.7. LEMMA. Let S be an ac-parasemifield. Then 2 ⭈ 1 S FS a q ay1 q Ž n ⭈ 1 S .y1 for e¨ ery a g S and e¨ ery positi¨ e integer n. Proof. For positive integers m, n put f Ž m. s 2 m and bn s Ž2 n y 1.1 S ⭈ Ž n ⭈ 1 S .y1 g S. Now, if n G 1, then Žusing standard methods. we can find m G 1 such that

Ž Ž 2 n y 1 . rn . f

Ž m.

F

ž

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/

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On the other hand, using the binomial formula for Ž a q ay1 . f Ž m., we also find that

ž

f Ž m. f Ž m y 1.

/

⭈ 1 S FS Ž a q ay1 .

f Ž m.

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Consequently, bnf Ž m. FS Ž a q ay1 . f Ž m. and, by 4.6, we have bn FS a q ay1. The semiring S is said to be archimedean if for all a, b g S there exists a positive integer n such that b FS na Žequivalently, for all a, b g S, there exist c g S and a positive integer m such that b q c s ma.. Note that if this is true, then for all x g R and a g S there exists n G 1 with x FS na. 4.8. LEMMA.

Let S be an archimedean conical ac-parasemifield. Then

Ži. 2 ⭈ 1 S FS a q ay1 for e¨ ery a g S. Žii. 2 ab FS a2 q b 2 for all a, b g S. Proof. By 4.2Žv., there exists a field F such that S is a subsemiring of F and F s  a y b; a, b g S4 . Clearly, the characteristic of F is zero and hence we can assume that ⺡ Žthe field of rationals. is the prime subfield of F. Let n be a positive integer and a g S, a / 1. By 4.7, a q ay1 s Ž2 n y 1.rn q c n for some c n g S. Put ¨ s a q ay1 y 2 g F, ¨ / 0, since a / 1. Then ¨ q 1rn s c n , n¨ q 1 s nc n g S. Further, y¨ y1 g F and y¨ y1 s d y e, d, e g S, e s d q ¨ y1 . Since S is archimedean, we have d FS m for a suitable positive integer m, m s d q f, f g S. Now, ¨ y1 q m s ¨ y1 q d q f s e q f g S, 1 q ¨ m s ¨ Ž e q f .. As we have already proved, 1 q ¨ m g S, and so ¨ s Ž1 q ¨ m.Ž e q f .y1 g S. Thus a q ay1 s ¨ q 2 and 2 FS a q ay1 . Finally, let a, b g S, a / b, w s a y b g F. Then w 2 ay1 by1 q bay1 y 2 g S by the preceding part of the proof.

5. CONGRUENCE-SIMPLE MULTIPLICATIVELY CANCELLATIVE ADDITIVELY IDEMPOTENT COMMUTATIVE SEMIRINGS 5.1. EXAMPLE. Let A be a non-zero subsemigroup of the additive group ⺢Žq. of real numbers. We shall consider the following Žcommutative. mcai-semiring W s W Ž A. s W Ž[, ).: W s A, a [ b s minŽ a, b ., and a) b s a q b for all a, b g A. 5.1.1. LEMMA. W is congruence-simple if and only if A l ⺢q/ ⭋ / A l ⺢y.

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Proof. First, assume that W is congruence-simple. If A l ⺢ys ⭋, 0 / a g A, and if a relation r is defined on W by Ž x, y . g r if and only if either x s y or 2 a F minŽ x, y ., then r is a non-trivial congruence of W, a contradiction. We proceed similarly if A l ⺢qs ⭋. Now, assume that A l ⺢q/ ⭋ / A l ⺢y and let r / id W be a congruence of W. Take a, b g W; a - b; and c g A l ⺢q. Since r / id W and A l ⺢y/ ⭋, there are e, f g W such that Ž e, f . g r, e - f F a, and c q b y a F f y e. Let n be the greatest non-negative integer such that e q nc F a. Then Ž n q 1. c q e q b y a F f q nc, 0 - Ž n q 1. c q e y a F f q nc y b, b - f q nc. Of course, Ž e q nc, f q nc . g r, Ž e q nc . [ a s e q nc, Ž f q nc . [ a s a, Ž e q nc, a. g r. Quite similarly, Ž e q nc, b . g r and Ž a, b . g r. Thus r s W = W. 5.1.2. LEMMA. Ži. W is ideal-simple if and only if A is a subgroup of ⺢Žq. Ž i.e., W is a parasemifield.. Žii. W is finitely generated if and only if so is the semigroup AŽq.. 5.1.3. LEMMA. Ži. W Ž[. possesses a neutral element if and only if A l ⺢qs ⭋ and supŽ A. g A. Žii. W Ž[. possesses an absorbing element if and only if A l ⺢ys ⭋ and infŽ A. g A. Žiii. W Ž). possesses a neutral element if and only if 0 g A. Živ. W Ž). does not possess an absorbing element. 5.1.4. LEMMA. If A and B are subsemigroups of ⺢Žq. such that A l ⺢q / ⭋ / B l ⺢q and A l ⺢y/ ⭋ / B l ⺢y, then W Ž A. ( W Ž B . if and only if B s qA for some q g ⺢q. 5.2. PROPOSITION. Let S be a cg-simple ai-parasemifield. Then the semilattice SŽq. is a chain, i.e., a q b g  a, b4 for all a, b g S. Proof. The relation F defined on S by x F y if and only if x q y s x is an ordering Žcompatible with addition and multiplication. and we have to show that this ordering is linear. Anyway, note first that since SŽ⭈. is a group, SŽq. contains no smallest Ži.e., absorbing. element and also no greatest Ži.e., neutral . element. Now, take w g S and put T s  a g S; aw F w4 . Then 1 g T and T is a subsemiring of S. Moreover, Tx l Ty s T Ž x q y . for all x, y g S. Indeed, if a, b g T and ax s by s ¨ , then ¨ Ž x q y .y1 w F wx Ž x q y .y1 , ¨ Ž x q y .y1 w F wy Ž x q y .y1 , ¨ Ž x q y .y1 w F w Ž x q y .Ž x q y .y1 s w, ¨ s ¨ Ž x q y .y1 Ž x q y . g T Ž x q y .. Conversely, if c g T, then cŽ x q y . xy1 w s cw q cyxy1 w F w, cŽ x q y . g Tx, and, similarly, cŽ x q y . g Ty. Now, defining a relation r on S by Ž x, y . g r if and only if Tx s Ty, we get a congruence of the semiring S. If r s S = S, then T s S, aw F w for

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every a g S, and w is a greatest element of S, a contradiction. Thus r s id S and x ¬ Tx is an injective mapping. Let x, y g S. If x F y and a g S, then yy1 F xy1 , axyy1 w F axxy1 w s aw F w, axyy1 g T, and we see that Tx : Ty. Conversely, if Tx : Ty, then Tx s Tx l Ty s T Ž x q y ., and hence x s x q y, x F y. Then x F y if and only if Tx : Ty and our aim is to show that the set Tx; x g S4 is linearly ordered by inclusion. As one may see easily, this is true if and only if T j Ty1 s S and now, proceeding by contradiction, we shall assume that a g S _ŽT j Ty1 .. For every n G 0, let Pn s T l Ta l ⭈⭈⭈ l Ta n s T Ž1 q a q ⭈⭈⭈ qa n . and, for x g S, let Q x denote the set of u g S such that Pm : Txu for at least on m G 0. Then s a is a congruence of S, where Ž x, y . g s a if and only if Q x s Q y , and we shall first assume that s a s id S . Clearly, Q1q a : Q1. Conversely, if Pm : Tu, then Pmq1 : Pm a : Tau, Pmq 1 : Pm : Tu, Pmq1 : Tau l Tu s T Ž1 q a. u, and u g Q1qa . Thus, Ž1 q a, 1. g s a and consequently 1 q a s 1. However, then T s T Ž1 q a. s T l Ta : Ta and a g Ty1 , a contradiction. We have proved that s a s S = S, and therefore Ž x, 1. g s a for every x g S and there exists m G 0 such that Pm : Tx. Proceeding similarly Ži.e., replacing a by ay1 ., we can show that R n : Tx for some n G 0, R n s T l Tay1 l ⭈⭈⭈ l Tayn s T Ž1 q ay1 q ⭈⭈⭈ qayn .. In particular, Pk : Tay1 for some k G 0 and we have Pk a : T l Pk a s Pkq1 : Pk , Pk : Pk ay1 , and, by induction, Pk : Pk ayi : Tayi for every i G 0. Consequently, Pk : Tx, i.e., Tb : Tx, b s 1 q a q ⭈⭈⭈ qa k , for every x g S. In this way, we have shown that b is a smallest element of S and this is the needed final contradiction. 5.3. THEOREM. Let S be a congruence-simple multiplicati¨ ely cancellati¨ e additi¨ ely idempotent semiring Ž see 3.3.. The there exists a subsemigroup A of the additi¨ e group ⺢Žq. of real numbers such that A l ⺢q/ ⭋ / A l ⺢y and the semirings S and W Ž A. are isomorphic Ž see Example 5.1.. Proof. First, assume that S is a parasemifield. By Proposition 5.2, the semilattice SŽq. is a chain Žpossessing no smallest and no greatest element., and so SŽ⭈. is a linearly ordered Žabelian . group Ž x F y if and only if x q y s x .. We are going to make it clear that this order is archimedean. Let w g S, 1 - w. Define a relation t w on S by Ž x, y . g t if and only if either x s y or ¨ w m F x F ¨ w n and ¨ w m F y F ¨ w n for an element ¨ g S and some positive integers m, n, m F n. Then t w is a congruence of S and we have t w s S = S. In particular, given x g S, we have ¨ w m F x F ¨ w n, ¨ w m F w F ¨ w n, ¨ F w 1y m and x F w 1q nym . We have proved that SŽ⭈, F. is an archimedean linearly ordered Žabelian . Ž ᎐Baer᎐Cartan᎐Loonstra. group. According to the well-known Holder ¨

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Theorem Žw19; see also w2, 4, 24x., there exists an injective homomorphism ␸ of SŽ⭈. into the additive group ⺢Žq. such that x F y in SŽF. implies ␸ Ž x . F ␸ Ž y . in ⺢ŽF.. ŽIf, among the positive elements of SŽ⭈, F., there exists a smallest one, say w, then SŽ⭈. is an infinite cyclic group generated by w and we put ␸ Ž w n . s n. In the opposite case, we choose any w ) 1 and, for every x g S, we put ␸ Ž x . s sup mrn; m, n g ⺪, n ) 0, w m F x n4 . Consequently, S ( W Ž ␸ Ž S Ž⭈.... The general case now follows from an easy combination of Proposition 4.1, the preceding part of the proof, and Example 5.1. 6. A FEW CONSEQUENCES 6.1. COROLLARY Žcf. w26, 3.2x.. Let S be a congruence-simple semiring such that the additi¨ e semigroup SŽq. possesses a neutral element. Then just one of the following three cases takes place: Ž1. S is isomorphic to one of the two-element semirings Z3 , Z4 , Z5 , and Z6 ; Ž2. S is a zero-multiplication ring of finite prime order; Ž3. S is a field. Proof. Combine 3.1, 3.3, 5.3, 5.1.3, and 3.4. 6.2. COROLLARY Žcf. w26, 4.7x.. Let S be a congruence-simple semiring such that the multiplicati¨ e semigroup SŽ⭈. possesses an absorbing element. Then just one of the following two cases takes place: Ž1. S is isomorphic to one of the two-element semirings Z1 , Z2 , Z3 , Z4 , and Z5 . Ž2. There exists an abelian group G such that S is isomorphic to the semiring V Ž G . Ž see 3.2.. Proof. Combine 3.1, 3.3, 5.3, and 5.1.3. 6.3. COROLLARY. Let S be a congruence-simple semiring such that the multiplicati¨ e semigroup SŽ⭈. possesses an absorbing element. Then just one of the following four cases takes place: Ž1. and Z5 . Ž2. semiring Ž3. Ž4.

S is isomorphic to one of the two-element semirings Z1 , Z2 , Z3 , Z4 , There exists an abelian group G such that S is isomorphic to the V Ž G . Ž see 3.2.. S is a zero-multiplication ring of finite prime order. S is a field.

Proof. Combine 3.1, 3.3, 5.3, 5.1.3, and 3.4.

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6.4. Remark. Let S be a congruence-simple semiring such that SŽq. contains a neutral Žresp. absorbing. element 0 Žresp. o .. Then S0 s 0 Žresp. So s o . in all cases with the exception of the two-element semirings Z3 , Z6 Žresp. Z4 , Z5 .. 7. SEMIRINGS OF POSITIVE REAL NUMBERS The results of this section are of auxiliary character and to some extent are folklore. Anyway, for the benefit of the reader, full details are given. 7.1 Let F be a field containing as a Žprime. subfield the field ⺡ of rationals and let ᑛ denote the set of subsemirings S of F such that 0 f S; we have ⺡qg ᑛ and ᑛ is ordered by inclusion. Further, let ᑦ Žresp. ᑝ . denote the set of S g ᑛ such that 1 g S Žresp. ⺡q: S ., and for every a g S, let there be a positive integer n Žresp., positive integers p, q g ⺡q. such that a FS n Žresp., p FS a FS q .. Finally, let ᑧ denote the set of S g ᑛ such that F s  a y b; a, b g S4 . 7.1.1. LEMMA. Ži. E¨ ery S g ᑝ is archimedean. Žii. Žiii.

If S g ᑦ and SŽ⭈. is a group, then S is archimedean and S g ᑝ. E¨ ery S g ᑧ is conical.

Proof. Ži. If a, b g S, then p FS a and b FS q for some p, q g ⺡q and we get b FS qpy1 a. Žii. If a, b g S, then bay1 FS n and b FS na. Žiii. This is obvious. 7.1.2. LEMMA.

Let S g ᑧ and T g ᑛ be such that S : T. Then:

Ži. T g ᑧ. Žii. If S g ᑦ, then T g ᑦ. Žiii. If S g ᑦ and ⺡q: S, then T g ᑝ Ž in particular, S g ᑝ .. Proof. Ži. This is obvious. Žii. If u g T, then u s a y b; a, b g S; a FS n; and u s a y b FT n. Žiii. Let w g F; w / 0; wy1 s a y b; a, b g S; a -S n; n s a q c, c g S. Now, for d s b q c g S, we have wy1 q d s n; ny1 wy1 q ny1 d s 1; ny1 q ny1 dw s w; ny1 d g S; and hence ny1 FT w, provided that w g T. 7.1.3. LEMMA. Let S g ᑯ and T s  pa q q; p, q g ⺡qj 04 , p q q / 0, a g S4 . Then T is a subsemiring of F; S j ⺡q: T ; and T g ᑛ. Moreo¨ er, if S g ᑦ Ž resp., ᑝ, ᑧ ., then T g ᑦ Ž resp., ᑝ, ᑧ ..

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Proof. If pa q q s 0 for p q q / 0, then p / 0 / q and there is a positive integer n such that yn g S. Now, Žyn. 2 s n2 g S and n.Žyn. s yn2 g S, a contradiction. The rest is clear. 7.1.4. LEMMA. Let S g ᑛ, w g F, w 2 g S, and T s  a q bw; a, b g S j  04 , a q b / 04 . Then: Ži. T is a subsemiring of F and S : T Ž if 1 g S, then w g T .. Žii. If 0 g T, then yw s aby1 for some a, b g S. Žiii. If SŽ⭈. is a group and yw f S, then T g ᑛ. 7.1.5. LEMMA. Let S g ᑛ; w g F; w 2 g S; T1 s  a q bw; a, b g S j  04 , a q b / 04 ; and T2 s  a y bw; a, b g S j  04 , a q b / 04 . Then both T1 and T2 are subsemirings of F, S : T1 l T2 and either T1 g ᑛ or T2 g ᑛ. Proof. See the proof of Lemma 4.4. 7.1.6. LEMMA. Let S g ᑛ be such that 1 g S, and let u g S. Denote by T the set of all elements from F of the form a0 q a1 uy1 q ⭈⭈⭈ qa n uyn , where n G 0, a i g S j  04 , and a s Ýa i / 0 Ž then a g S .. Then T is a subsemiring of F, S j  uy1 4 : T, and T g ᑛ. Moreo¨ er, if S g ᑝ, then T g ᑝ. Proof. If a0 q a1 uy1 q ⭈⭈⭈ qa n uyn s 0, then a0 u n q a1 u ny1 q ⭈⭈⭈ qa ny 1 u q a n s 0, a contradiction with 0 f S. Thus T g ᑛ and, moreover, if p FS u FS q then qy1 FT F uy1 FT py1 , and the rest is clear. 7.1.7. LEMMA. Let S g ᑛ and T s  aby1 ; a, b g S4 . Then T g ᑛ and T Ž⭈. is a group Ž i.e., T is a parasemifield.. Moreo¨ er, if S is archimedean, then T is archimedean and T g ᑝ. Proof. The proof is easy. 7.1.8. PROPOSITION.

Let S g ᑦ l ᑧ be maximal in ᑦ l ᑧ. Then:

Ži. SŽ⭈. is a group and S is an archimedean conical parasemifield. Žii. For e¨ ery w g F, w / 0, we ha¨ e w 2 g S and either w g F or yw g F. Žiii. FS is linear Ž both on S and F .. Proof. By Lemma 7.1.3, ⺡q: S and, by 7.1.2Žiii., S g ᑝ. Further, by 7.1.6., SŽ⭈. is a group and by 7.1.1, S is both archimedean and conical. Now, w 2 g S by 4.8Žii. and we shall consider the semirings T1 , T2 defined in 7.1.5. We can assume that 0 f T1 s  a q bw; a, b g S j  04 , a q b / 04 , the other case being similar. Then T g ᑧ, S : T, and w g T. Finally, by 7.1.2Žiii., T g ᑝ l ᑧ and, due to the maximality of S, we must have T s S and w g S. 7.2. Let F be a field and let S be a subsemiring of F such that S g ᑦ l ᑧ; then charŽ F . s 0 and we may assume that ⺡ is a subfield of

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F. Now, let T g ᑦ l ᑧ be such that S : T and T is maximal in ᑦ l ᑧ. By 7.1.8 the ordering FT corresponding to T is an archimedean linear order on the field F. Now, by the Holder Theorem Žw19x; see the proof of ¨ 5.3. there exists an injective homomorphism ␸ : F Žq. ¬ ⺢Žq. such that ␸ Ž x . F ␸ Ž y . in the usual order of ⺢ whenever x, y g F and x FT y. Further Žsee w1x and w4x., for every a g F there is ␣ Ž a. g ⺢ such that ␸ Ž ax . s ␸ Ž x . ␣ Ž a. for every x g F; we have ␣ Ž0. s 0, ␣ Ž a. ) 0 for a )T 0, and ␣ Ž a q b . s ␣ Ž a. q ␣ Ž b . for all a, b g F. Consequently, we can find q g ⺢, q ) 0, such that ␣ Ž a. s ␸ Ž a. q for every a g F. Thus ␸ Ž xy . s ␸ Ž x . ␸ Ž y . q for all x, y g F, and the mapping ␺ : F ª ⺢, ␺ Ž x . s ␸ Ž x . q for every x g F is an Žinjective. homomorphism of the field F into the field ⺢; for x FT y in F we have ␺ Ž x . F ␺ Ž y . in ⺢. In particular, ␺ Ž S . : ␺ ŽT . : ⺢q. 7.3 Let F be a subfield of the field ⺢ of real numbers and let P Ž F . s  a12 q ⭈⭈⭈ qa2n ; n G 1, a i g F, a i / 04 . 7.3.1. LEMMA. Ži.

P Ž F . is a conical parasemifield.

Žii. F s  a y b; a, b g P Ž F .4 . Žiii. ⺡q: P Ž F . : Fq. Proof. If a s a12 q ⭈⭈⭈ qa2n , then ay1 s Ž a1 ay1 . 2 q ⭈⭈⭈ qŽ a n ay1 . 2 g P Ž F .. Further, x s Ž xr2. 2 q 1 y Ž xr2 y 1. 2 for every x g F. 7.3.2. LEMMA. parasemifield.

Fqs  a g F; a ) 04 is an archimedean and conical

7.3.3. LEMMA. Let S be a subsemiring of F such that P Ž F . : S : Fq. Then S is a conical parasemifield. Proof. We have ay1 s a.ay2 g S for every a g S. 7.3.4. LEMMA. Let S, T be subsemirings of F such that P Ž F . : S : Fq and P Ž F . : T : Fq. Then: Ži. Žii. Žiii.

S and T are conical. If S : T and S is archimedean, then T is also. If S, T are archimedean, then S l T is also.

7.3.5. LEMMA. Let S be a subsemiring of F such that 0 f S and 1 g S. Denote by AŽ S . the set of a g S such that n y a g S for some positi¨ e integer n. Then AŽ S . is a subsemiring of S, 1 g AŽ S ., and there is a positi¨ e integer m with m y a g AŽ S .. 7.3.6. EXAMPLE. Let F s ⺡ and S s  q g F; q G 14 . Then S is a subsemiring of F, S is both archimedean and conical, and S q S / S.

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7.3.7. EXAMPLE. Let F s ⺡Ž'2 .. Then P Ž F . / Fq Žy1 q '2 g F _ P Ž F .. and P Ž F . is an archimedean and conical parasemifield ŽŽ a q b'2 . 2 q Ž a y b'2 . 2 s 2 a2 q 4 b 2 and Lemma 7.3.5 applies.. q

7.3.8. EXAMPLE. Let ␣ g ⺢q be a transcendental number and F s Ž ⺡ ␣ .. Then P Ž F . is a conical parasemifield but P Ž F . is not archimedean Žwe have n y ␣y2 f P Ž F . for every positive integer n; to show this, just watch the behavior of f 2 Ž r . gy2 Ž r ., where f, g g ⺡w x x, f / 0 / g, and r g ⺢, r ª 0.. Further, for every a g P Ž F ., choose a positive integer m a such that m a y a g Fq and let S be the subsemiring of Fq generated by P Ž F . j  m a y a; a g P Ž F .4 . Then S is both archimedean and conical. 7.3.9. EXAMPLE. If F s ⺡, ⺢ or if F is the field of algebraic real numbers, then P Ž F . s Fq. 7.3.10. EXAMPLE. Let ␣ g ⺢q be a transcendental number, 0 - ␣ - 1, F s ⺡Ž ␣ ., and let ␸ : ⺡Ž x . ª F be the isomorphism such that ␸ Ž x . s ␣ and ␸ <⺡ s id. Now, denote by T the set of f g ⺡Ž x . such that there exist m, n g ⺞ with ny1 F f Ž u. F m for every u g ⺢, 0 F u F 1. Then S s ␸ ŽT . is a subsemiring of Fq and S is an archimedean parasemifield. On the other hand, S is not conical and P Ž F . ­ S. 7.4. LEMMA. Let S, T be subsemirings of ⺢ such that S : T, S is conical, and e¨ ery element of T is algebraic o¨ er the difference field F s S y S Žs RŽ S ... Then T is conical. Proof. Put R s T y T and take 0 / r g R. Then r s a y b, a, b g T, and there is a polynomial f g F w x, y x such that ry1 s f Ž a, b .. Since F s S y S, we conclude easily that ry1 g R. Thus R is a field and T is conical. 7.5. LEMMA. Let S and T be subsemirings of ⺢q such that both S and T are semisubtracti¨ e Ž see 4.5., ⺡q: S l T, and there exists a Ž semiring . isomorphism ␸ : S ª T. Then S s T and ␸ s id. Proof. Assume, on the contrary, that ␸ Ž a. - a for some a g S. Now, take q g ⺡q such that ␸ Ž a. - q - a. Then a y q g S and 0 - ␸ Ž a y q . s ␸ Ž a. y q - 0, a contradiction. 7.6. Remark. Let F be a field such that 0 f P Ž F . s  a12 q ⭈⭈⭈ qa2n ; n G 1, 0 / a i g F 4 Ž P Ž F . is a conical parasemifield and F s P Ž F . y P Ž F .., and let ᑛ denote the set of subsemirings S of F such that P Ž F . : S and 0 f S. Again, every S g ᑛ is a conical parasemifield and S is maximal in ᑛ if and only if S g ᑧ, where ᑧ is the set of semisubtractive parasemifields from ᑛ.

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Take S g ᑛ and w g F such that w f S and yw f S. Then S1 s  a y bw; a, b g S j  04 , a q b / 04 g ᑛ; S : S1; yw g S1; and S1 : T, where T is maximal in ᑛ; we have T g ᑧ and w f T. Now, it follows easily that S s F ᑧ S , where ᑧ S s T g ᑧ; S : T 4 . 8. CONGRUENCE-SIMPLE MULTIPLICATIVELY AND ADDITIVELY CANCELLATIVE COMMUTATIVE SEMIRINGS 8.1. LEMMA.

Let S be a cg-simple amc-semiring. Then S is archimedean.

Proof. Define a relation r on S by Ž a, b . g r if and only if there exist c, d g S and m, n g ⺞ such that a q c s mb and b q d s na. It is easy to check that r is a congruence of S such that Ž a, 2 a. g r for every a g S. Since a / 2 a, we have r s S = S. 8.2. THEOREM. Let S be a non-tri¨ ial amc-semiring. Then S is congruence-simple if and only if S satisfies the following three conditions: Ž1. For all a, b g S there exist c g S and a positi¨ e integer n such that b q c s na Ž i.e., S is archimedean.. Ž2. For all a, b, c, d g S, a / b, there exist e, f g S such that ae q bf q c s af q be q d Ž i.e., S is conical .. Ž3. For all a, b g S there exist c, d g S such that bc q d s a Ž i.e., S is bi-ideal-simple.. Proof. If S is cg-simple, then Ž1. is satisfied by 8.1, Ž2. by 4.3Žiii., and 4.2Žv., and Ž3. follows from the easy fact that Sb q S is a bi-ideal of S. Now, conversely, assume that the conditions Ž1., Ž2., and Ž3. are satisfied. The rest of the proof is divided into four parts: Ži. Let r / id S be a congruence of S such that the corresponding factor-semiring T s Srr is additively cancellative. The difference ring R s RŽ S . of S is a field Žby Žii. and 4.2Žv.. and the set I s  a y b; Ž a, b . g r 4 is a non-zero ideal of R. Consequently, I s R and for all x, y g S there are a, b g S such that Ž a, b . g r and x y y s a y b. Now, x q b s y q a and Ž x, y . g r, since T is an ac-semiring. We have proved that r s S = S. Žii. Let T be a non-trivial semiring satisfying the conditions Ž1., Ž3. Žand Ž2.. and let w g T be such that w q w s w. We claim that T is a ring Žfield.. First, according to Ž1., for every x g T there exists y g T such that x q y s w. In particular, if x q x s x, then w q x s x q y q x s x q y s w. On the other hand, replacing w by x, we get x q w s x. Thus x s w is

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the only idempotent of the additive semigroup T Žq. and it follows easily that Tw s w and, by Ž3., w q T s T. Consequently, w s 0 is a neutral element of T Žq. and we conclude that T Žq. is an Žabelian . group and T is a ring. Finally, if T satisfies Ž2., a g T, a / 0, and c g T, then ae q c s ae q 0 f q c s af q 0 e q 0 s af and c s aŽ e y f . for some e, f g T. Thus T is a field. Žiii. Let r be a congruence of S such that Ž w, 2 w . g r for at least one w g S. We claim that r s S = S. Assume on the contrary, that r / S = S and put T s Srr. By Žii., T is a ring, and hence r s id S by Ži., a contradiction, since S is not a ring. Živ. Let s be a congruence of S, s / S = S, and let w g S be arbitrary. By Žiii., Ž w, 2 w . f s and we can consider a congruence r of S maximal with respect to s : r and Ž w, 2 w . f r. Now, again by Žiii., r is a maximal congruence of S and T s Srr is a cg-simple semiring such that T Žq. has no idempotent. Using 3.1 we see that T is an ac-semiring, and so r s s s id S by Ži.. Thus S is cg-simple. 8.3. COROLLARY. Let S be an ac-parasemifield. Then S is cg-simple if and only if the following two conditions are satisfied: Ž1. For e¨ ery a g S there exist b g S and a positi¨ e integer n such that a q b s n1 S Ž i.e., S is archimedean.. Ž2. For e¨ ery a g S, a / 1 S , there exist b, c g S such that a q b s 1 S q ab q c Ž i.e., S is conical .. 8.4. Remark. Let S be a non-trivial amc-semiring such that 1 S g S. Then S is cg-simple if and only if S is both archimedean and conical and, moreover, S q S s S Ži.e., 1 S s a q b for some a, b g S .. To show this, we can proceed similarly as in the proof of 8.2. Note that Ž1 S , 2 S . f r for every congruence r / S = S and, if r is maximal with respect to Ž1 S , 2 S . f r, then r is a maximal congruence of S. 8.5. THEOREM. Let S be a congruence-simple additi¨ ely and multiplicati¨ ely cancellati¨ e semiring and let F s RŽ S . Ž see 4.2.. Then F is a field and there exists an Ž injecti¨ e . homomorphism ␸ of the field F into the field ⺢ of real numbers such that ␸ Ž S . : ⺢q. Moreo¨ er, if S is a parasemifield, then P Ž F . s  a12 q ⭈⭈⭈ qa2n ; 0 / a i g F, n G 14 : S. Proof. In view of 4.1, we may assume that S is a parasemifield. By Proposition 4.3, S is conical, and hence F is a field. Further, by Lemma 8.1, S is archimedean and the rest is clear from 4.8 and 7.2. 8.6. THEOREM w21, 6.9; 10, Satz 16x. Let E be a subfield of ⺢ and S be a subsemiring of ⺢ such that S is a parasemifield, Eq: S, and e¨ ery element

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from S is algebraic o¨ er E. Then S is congruence-simple Ž i.e., S is archimedean and conical .. 8.7. COROLLARY w23, Satz 2x. Let S be a subsemiring of ⺢ such that S is a parasemifield and e¨ ery element from S is algebraic o¨ er ⺡. Then S is congruence-simple. 8.8. COROLLARY Žcf. 7.3.8.. Let F be a subfield of ⺢ such that F is algebraic o¨ er ⺡. Then P Ž F . s  a12 q ⭈⭈⭈ qa2n ; 0 / a i g F 4 is a congruencesimple parasemifield Ž in particular, P Ž F . is achimedean.. 8.9. Remark. Let E, F be subfields of ⺢ such that E : F and the linear dimension dimŽ FE . is finite. Denote by ᑧ the set of subsemirings S of F such that F s S y S, S is a parasemifield, and Eq: S. Ži. We have F s E Ž w . for some w g Fq and we denote by Ž w s. w 1 , w 2 , . . . , wn , n G 1, all Žpair-wise different. real roots of the minimal polynomial of w over E. Now, there exist exactly n different homomorphisms ␸ 1 , . . . , ␸n : F ª ⺢ such that ␸ < E s id and we may assume that Ž⺢q. s ␸y1 Ž E Ž wi .q. . Clearly, Ti are ␸ 1 s id and ␸ i Ž w . s wi . Let Ti s ␸y1 i i q semisubtractive parasemifields, T1 s F , Ti g ᑧ, Ti are maximal in ᑧ, and P Ž F . : Ti . Žii. Let TM designate the intersection Fi g M Ti for every non-empty subset M of N s  1, 2, . . . , n4 . By w10, Satz 16x, TM 1 / TM 2 for M1 / M2 and ᑧ s TM ; ⭋ / M : N 4 . Consequently, ᑧ contains just 2 n y 1 elements and TN s T1 F ⭈⭈⭈ FTn is a smallest element of ᑧ. Clearly, P Ž F . : TN s  a12 b1 q ⭈⭈⭈ qa2m bm ; m G 1, 0 / a i g F, bi g Eq 4 . Note also that if S is a subsemiring of F such that TN : S and 0 f S, then S is a parasemifield and S g ᑧ, so that S s TM for some ⭋ / M : N. Similarly, if S is a subsemiring of F such that TN : S and 0 g S, then either S s TM j  04 or S s F. Žiii. If Eq: P Ž F . Že.g., E s ⺡., then TN s P Ž F . and ᑧ is the set of subsemirings S of F such that F s S y S Žs RŽ S .. and S is a parasemifield. Živ. Žcf. 7.5.. Let M1 and M2 be non-empty parts of N such that there exists an isomorphism ␸ : TM 1:ª TM 2 of the parasemifields with ␸ < Eqs id. Then ␸ extends to an E-automorphism of F, and therefore there is k g N such that w k g F and ␸ s ␸ k < TM 1. Further, there is a Ž . permutation p of N such that ␸ i ␸y1 k s ␸ pŽ i. and ␸ k Ti TpŽ i. for every i g N. Now, if M1 s  i1 , i 2 , . . . , i m 4 , then M2 s  pŽ i1 ., pŽ i 2 ., . . . , pŽ i m .4 ; in particular, the sets M1 and M2 have the same number of elements. 8.10. Remark. Let E, F be subfields of ⺢ such that E : F and F be algebraic over E. Denote by ᑧ the set of subsemirings of F such that

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F s S y S, S is a parasemifield, and Eq: S, and denote by ⌽ the set of homomorphisms ␸ : F ª ⺢ such that ␸ < E s id. Let T␸ s ␸y1 Ž⺢q. for every ␸ g ⌽. Clearly, T␸ are subsemisubtractive parasemifields, Tid s Fq, T␸ g ᑧ, T␸ are maximal in ᑧ, and P Ž F . : T␸ . Let TM designate the intersection F␸ g M T␸ for every non-empty subset M of ⌽. Clearly, TM g ᑧ and P Ž F . : TM . Now, we show that ᑧ s TM ; ⭋ / M : ⌽4 . Let S g ᑧ, a, b g S, and F1 s E Ž ay1 b ., S1 s S l F1. Then Eq: S1 and S1 is archimedean by w10, Satz 16x. In particular, n y ay1 b g S1 and na y b g S for some n g ⺞. We have shown that S is archimedean, and therefore P Ž F . : S Ž4.8Žii... Now, it follows from 4.5 that S s TM for some ⭋ / M : ⌽. 8.11. THEOREM Žcf. 9.7.. Let S be an archimedean and conical amc-semiring such that S has a unit element 1 S and Ž n1 S .y1 g S for e¨ ery n g ⺞. Then S is a parasemifield. Proof. We make use of a variation on the well-known Goodearl᎐ Handelman method Žsee w13x for details and further references ., the backgrounds of which go back to Holder and Hilbert Žw19x and w18x; but see ¨ w x. also 1᎐9, 15, 22, 24, 29᎐31 . Since S is conical, there exists a field F such that S is a subsemiring of F s S y S Žsee 4.2Ži, v.. and, since S is archimedean, we have S : A, where A denotes the set of a g F such that for every x g F there is n g ⺞ with na y x g S. Now, an Žadditive. homomorphism f : F Žq. ª ⺢Žq. will be called normative if f Ž1 F . s 1 and f Ž a. ) 0 for every a g S. The set N of normative homomorphisms is a compact convex set and we denote by E the set of extremal points of N . Now, one can show that every f g E is a ring homomorphism; see e.g. w14x Žin fact, no topological arguments are necessary and a rather easy and purely algebraic proof of the mentioned fact is availableᎏthe kind reader may wish to make it up as a stimulating exercise.. The rest of the present proof is divided into two parts: Ži. First, we check that a g A if and only if f Ž a. ) 0 for every normative ring homomorphism f : F ª ⺢. Indeed, if a g A, then na y 1 F g S for some n g ⺞ and we have f Ž a. G ny1 ) 0. To show the converse implication, put r s sup nrm; n g ⺪, m g ⺞, ma y n1 F g S j  044 . Then there exists f g E such that f Ž a. s r, and therefore f ) 0 and ma y n1 F g S j  04 for some m, n g ⺞. Now, it follows easily that a g A. Žii. Now, according to Ži., AŽ⭈. is a subgroup of F Ž⭈., and A is a parasemifield, and it remains to show that A s S. But, if a g A, then na g S for some n g ⺞ and consequently a s ny1 ⭈ na g S.

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8.12. COROLLARY. Let S be a congruence-simple amc-semiring such that 1 S g S. Then S is a parasemifield if and only if Ž n1 S .y1 g S for e¨ ery n g ⺞. 8.13. COROLLARY. Let S be a congruence-simple amc-semiring such that 1 S g S. Then for e¨ ery a g S there exist b g S and n g ⺞ such that ab s n1 S . 8.14. EXAMPLE. Let ␣ g ⺢q be a transcendental number and F s Ž ⺡ ␣ .. There exists an isomorphism ␸ : ⺡Ž x . ª F such that ␸ Ž x . s ␣ and ␸ ¬ ⺡ s id. Now, we denote by S the set of a g F such that a g ⺢q and ␸ Ž a. g ⺢q, where ␸ Ž a. s ␸ ŽŽ ␸y1 Ž a..X . Žhere, for f g ⺡Ž x ., f X is the derivative of f .. Clearly, S is a subsemiring of Fq, S is both archimedean and conical, and ⺡qS : S. Moreover, given a, b g S, we find q g ⺡q small enough such that qa2 - b and q⭸ Ž a2 . - ⭸ Ž b .. That is, b y Ž qa. a g S and we see that S is bi-ideal-simple. But 8.2, S is a congruence-simple amc-semiring. Finally, note that S l ⺡ s ⭋; in particular, S is not a parasemifield.

9. SUBSEMIRINGS OF ⺡ q 9.1. LEMMA. A subsemiring S of ⺡q is archimedean and conical if and only if for e¨ ery n g ⺞ there exists m g ⺞ such that krn g S for e¨ ery k G m. Proof. Assume first that S is both archimedean and conical. For n g ⺞, we can find r, s g S l ⺞ such that Ž rn y 1.rn g S and ŽŽ rn y 1. s y 1.rŽ rn y 1. g S. Now, put a s nŽŽ rn y 1. s y 1. and b s Ž rn y 1. 2 . Then gcdŽ a, b . s 1, and hence there is m g ⺞ such that  m, m q 1, m q 2, . . . 4 :  ua q ¨ b; u, ¨ g ⺞4 . Since arnŽ rn y 1. g S and brnŽ rn y 1. g S, we have krnŽ rn y 1. g S for every k g m. Consequently, krn g S. Now, assume that the condition of the lemma is satisfied and let a, b, r, s g ⺞, arb g S. Then there is m such that krbs g S for every k G m. Taking l g ⺞ such that las y br G m, we get l ⭈ arb y rrs g S. 9.2. PROPOSITION. Let S be a subsemiring of ⺡q such that S is archimedean and conical and 1rr g S for some r g ⺞, r G 2. Then S s ⺡q. Proof. If n g ⺞, then r lrn g S for some l g ⺞ Žby 9.1., and therefore 1rn s Ž1rr l .Ž r lrn. g S. 9.3. LEMMA. Let a, b, c g ⺞ be such that a - b, c - b and gcdŽ a, c . s 1. Then 1rb g S, where S denotes the subsemiring of ⺡q generated by arb and crb. Proof. Let n g ⺞ be such that n G 2 and Ž n2 . G Ž n q 1.Ž b y 1. 4 . We will construct a sequence r 0 , . . . , rn of integers such that 0 F ri F c for

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every 0 F i F n. First, since gcdŽ a nq 1, c . s 1, there is 0 F r 0 - c such that b n ' r 0 a nq1 Žmod c .. Quite similarly, gcdŽ a n, c . s 1, Ž b n y r 0 a nq1 .rc ' r 1 a n Žmod c . for some 0 F r 1 - c, and b n ' Ž r 0 a nq 1 q r 1 a n c . Žmod c 2 .. Proceeding by induction, we find the remaining numbers r 2 , . . . , rn such that b n ' Ž r 0 a nq 1 q r 1 a n c q ⭈⭈⭈ qri a nq 1yi c i . Ž mod c iq1 . for every 0 F i F n. Now, put s s Ý nis0 ri a nq 1yi c i. Since a - b and c - b, we have s F Ý nis0 Ž b y 1.Ž b y 1. nq 1yi Ž b y 1. i s Ž n q 1.Ž b y 1. nq 2 F Ž n2 .Ž b y 1. ny 2 F Ž b y 1 q 1. n s b n, and therefore t s b n y s G 0. On the other hand, t s rnq 1 c nq 1 and b n s s q rnq1 c nq 1. Finally, it is clear from the definition of s that Ž s q rnq 1 c nq 1 .rb nq 1 g S and we have proved that 1rb g S. 9.4. LEMMA. Let a, b, c, d g ⺞ be such that a - b, c - d and gcdŽ a, b . s gcdŽ c, d . s gcdŽ a, c . s 1. Then 1rscmŽ b, d . g S, where S denotes the subsemiring of ⺡q generated by arb and crd. Proof. We have arb s erg, crd s frg, and gcdŽ e, f . s 1, where g s scmŽ b, d .. It remains to use 9.3. 9.5. THEOREM. Let S be a congruence-simple subsemiring of ⺡q such that 1 g S. Then S s ⺡q. Proof. By 8.2, S is both archimedean and conical and there exist a, b, c,d g ⺞ such that arb g S, crd g S, arb q crd s 1, and gcdŽ a, b . s 1 s gcdŽ c, d .. It remains to apply Lemma 9.4 and Proposition 9.2. 9.6. EXAMPLE. Every positive rational number q can Žuniquely. be written as q s 2 dŽ q. ⭈ rsy1, where d Ž q . g ⺪, r, s g ⺞, and gcdŽ r, s . s gcdŽ r, 2. s gcdŽ s, 2. s 1. Clearly, d Ž p q q . G minŽ d Ž p ., d Ž q .., d Ž pq . s d Ž p . q d Ž q . for all p, q g ⺡q, and we put < q < 2 s 2yd Ž q. Žthe dyadic norm.. Now, the set S s  q g ⺡q; q y < q < 2 ) 04 is a Žproper. subsemiring of ⺡q Že.g., 2, 3, 4, ⭈⭈⭈ g S, 2r3 g S, 1 f S, 1r2 f S . and, for every t g ⺡, one finds easily n g ⺞ such that n y t g S. Thus S is both archimedean and conical. Furthermore, if p, q g S and w s p y < p < 2 , then w g ⺡q and we take t g S such that tq F w and d Ž p . F d Ž tq .. Now, p y tq g S and we see that S is bi-ideal simple. By Theorem 8.2, S is a congruence-simple amc-semiring Ž⺡q ­ S and S is not a parasemifield.. 9.7. Remark. Using Proposition 9.2, we can improve Theorem 8.11 as follows: Let S be an archimedean and conical amc-semiring such that Ž n1 S .y1 g S for at least one n g ⺞, n G 2. We claim that S is a parasemifield. Indeed, with respect to 7.2 Žsee also the proof of 8.11. we may assume that S is a subsemiring of ⺢q. Now, F s S y S is a subfield of ⺢ and we

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put S1 s S l ⺡q. Then 1 g S1 , 1rn g S1 , and S1 is an archimedean and conical subsemiring of ⺡q. By Proposition 9.2, S1 s ⺡q, and S is a parasemifield.

10. CONGRUENCE-SIMPLE COMMUTATIVE SEMIRINGSᎏSUMMARY 10.1. THEOREM. A Ž commutati¨ e . semiring S is congruence-simple if and only if S is Ž isomorphic to . one of the following semirings: Ž1. Ž2.

the two-element semirings Z1 , Z2 , Z3 , Z4 , and Z5 Ž see Section 2.. the semirings V Ž G . for any abelian group G Ž see 3.2.;

Ž3. the semirings W Ž A. for any subsemigroup A of the additi¨ e group ⺢Žq. of real numbers such that A l ⺢q/ ⭋ / A l ⺢y Ž see Example 5.1.; Ž4. fields; Ž5. zero-multiplication rings of finite prime order; Ž6. the subsemirings S of the semiring ⺢q of positi¨ e real numbers such that the following three conditions are satisfied: Ž6a. for all a, b g S there exist c g S and a positi¨ e integer n such that b q c s na; Ž6b. for all a, b, c, d g S, a / b, there exist e, f g S such that ae q bf q c s af q be q d; Ž6c. for all a, b g S there exist c, d g S such that bc q d s a. Proof. Combine 3.1, 3.2, 3.4, 5.3, 8.3, and 8.6. 10.2. Remark. Ži. The two-element semirings Z1 , . . . , Z5 are pair-wise non-isomorphic. Žii. V Ž G 1 . ( V Ž G 2 . if and only if G 1 ( G 2 Žsee 3.2.. Žiii. W Ž A1 . ( W Ž A 2 . if and only if A 2 s qA1 for some q g ⺢q Žsee 5.1.4.. 10.3. Remark. The congruence-simple semirings of type 10.1Ž6. are not yet fully classified up to isomorphism Žsee 8.9 and 8.10 for some special cases.. In particular, the following problem Žoriginally formulated in w26, 5.7x. remains open: Does there exist a congruence-simple amc-semiring S such that 1 S g S and S is not a parasemifield? According to 9.5 and 9.7 we know that if S were such a semiring, then S ­ ⺡q and Ž n1 S .y1 f S for every n g ⺞, n G 2. Examples of cg-simple amc-semirings without unit are given in 8.14 and 9.6.

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11. IDEAL-SIMPLE COMMUTATIVE SEMIRINGSᎏ BASIC CLASSIFICATION Ideal-simple semirings are better known than the congruence-simple onesᎏsee e.g. w10᎐12, 16, 17, 20, 21, 23, 27, 28, 32, 34᎐41x. Anyway, for the sake of completeness and the full comfort of the reader, we include some basic information on ideal-simple Žcommutative. semirings. The reader is urged to compare the results on the congruence-simple semirings with those on the ideal-simple ones. A non-trivial Žcommutative. semiring S is said to be a semifield if there exists an element w g S such that Sw s w and T is a subgroup of SŽ⭈., where T s S _ w4 . If, moreover, S is not a field, then one says that S is a proper semifield. 11.1. PROPOSITION. Žii.

Ži.

E¨ ery semifield is ideal-simple.

E¨ ery parasemifield is ideal-simple.

11.2. THEOREM. Let S be an ideal-simple semiring. Then just one of the following fi¨ e cases takes place: Ž1. Ž2. Ž3. Ž4. Ž5.

S is isomorphic to one of the two-element semirings Z1 , Z3 , and Z4 ; S is a zero-multiplication ring of finite prime order; S is a field; S is a proper semifield; S is a parasemifield.

Proof. Suppose that S contains at least three elements and put A s  a g S; < Sa < s 14 . If b g S _ A, then Sb s S, and hence S is a parasemifield, provided that A s ⭋. Assume therefore that A / ⭋; then there is an element w g S such that SA s  w4 , w s w q w, and we see that A is an ideal of S. Consequently, either A s  w4 or A s S. First, let A s S and S q w s w Ži.e., w s o is an absorbing element of SŽq... Then SS s o and, if P is a subsemigroup of SŽq. such that o g S, then P is an ideal of S, and hence either P s  o4 or P s S. If a g S, a q a s a, then  a, o4 is a subsemigroup of SŽq.,  a, o4 / S, and therefore a s o. We have proved that w is the only idempotent element of SŽq.. Now, take a g S, a / o, and put P s  na; n G 14 . Then P j  o4 s S, P Žq. contains no proper subsemigroup, P Žq. is a finite cyclic semigroup, o g P, and there is b g P such that b / o and b q b s o. But then  b, o4 is a subsemigroup of SŽq., a contradiction with < S < G 3. Next, let A s S and S q w / w. Then S q w s S, since S q w is a Žbi-. ideal of S, and it follows that w s 0 is a neutral element of SŽq.. Further, proceeding similarly as in the preceding part of the proof, we can show

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that 0 is only the only idempotent of SŽq. and that every cyclic subsemigroup of SŽq. is finite. Consequently, SŽq. is a group and S is a ring. Thus either Ž2. or Ž3. takes place. Further, let A s  w4 and S q w s w, w s o. For every a g T s S _ o4 , the set Ia s  b g S; ab s o4 is an ideal of S, Ia / S, and so Ia s o. We have checked that T is a subsemigroup of SŽ⭈. and, since Ta s T for every a g T, T Ž⭈. is a group. Finally, let A s  w4 and S q w / w. Then S q w s S, w s 0 is a neutral element of SŽq., and, proceeding similarly as in the preceding part, we show that T s S _ 04 is a subgroup of SŽ⭈.. 12. SEMIFIELDS 12.1. PROPOSITION. Let S be a semifield and let w g S be such that Sw s w Ž and T s S _ w4 is a subgroup of SŽ⭈... Then just one of the following four cases takes place: Ž1. S is isomorphic to the two-element semiring Z2 Ž and then w s o is an absorbing element of both SŽq. and SŽ⭈..; Ž2. w s o is an absorbing element of both SŽq. and SŽ⭈., and S q a / o for e¨ ery a g T ; Ž3. S is a field Ž and then w s 0 is a neutral element of SŽq..; Ž4. w s 0 is a neutral element of SŽq. and S is a proper semifield Ž i.e., SŽq. is not a group.. Proof. We have w q w s Ž1 S q 1 S . w s w and, if 1 S q w s w, then w s aw s aŽ1 S q w . s a q w for every a g S, and so w s o. On the other hand, if 1 S q w / w, then 1 S s Ž1 S q w .y1 Ž1 q w . s 1 S q wy1 q w, 1 S q w s 1 S q wy1 q w q w s 1 S q wy1 q w s 1 S , and a q w s a1 S q aw s aŽ1 S q w . s a1 S s a for every a g S, and so a s 0. The rest is clear. 12.2. PROPOSITION. Let S be a semifield of type 12.1Ž2.. Define a relation ␳S on S by Ž a, b . g ␳S if and only if  x g S; a q x s o4 s  y g S; b q y s o4 . Then Ži. ␳S is a congruence of S and Ž a, o . f ␳S for e¨ ery a g T s S _ o4 . Žii. P s  a g T ; Ž a, 1 S . g ␳S 4 is a subgroup of T Ž⭈. and either P s 1 Ž and ␳S s id S . or P is a parasemifield. Žiii. The factor-semiring R s Sr␳S is isomorphic to the Ž additi¨ ely idempotent . semifield V ŽTrP . Ž see 3.2. and ␳ R s id R . Proof. The assertions Ži., Žii. and the fact that ␳ R s id R are easy to check. Further, let r be a congruence of the semiring R, r / id R . We have

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Ž x, y . g r for some x, y g R, x / y, and we put A s  r g R; Ž z, o . g R4 . Then A is an ideal of R and, since Ž x, y . f ␳ R s id R , we have A /  o4 . Thus A s R Žsince R is ideal-simple. and r s R = R. We have proved that R is a cg-simple semiring and the result follows from 3.1, 3.2, and 3.3. 12.3. Remark. Let P be a semiring such that either P is trivial or P is a parasemifield. Suppose further that P Ž⭈. is a subgroup of an Žabelian . group T Ž⭈., put S s T j  o4 , and define an addition on S as follows: Ža. Žb. Žc.

x q o s o s o q x for every x g S; x q y s o for all x, y g T, xy1 y f P; x q y s Ž1T q xy1 y . x s Ž1T q yy1 x . y for all x, y g T, xy1 y g P.

Then S becomes a semifield of type 12.1Ž2. and every semifield of that type may be constructed in the described way. 12.4. PROPOSITION.

Let S be a proper semifield of type 12.1Ž4.. Then:

Ži. a q b / 0 for all a, b g T s S _ 04 . Žii. T is a subsemiring of S and either T is tri¨ ial Ž and then S ( Z5 . or T is a parasemifield. Proof. Since SŽq. is not a group, the set T1 s  a g T ; 0 f S q a4 is non-empty. Now, bT1 s T1 for every b g T, and so T1 s T. The rest is clear. 12.5. Remark. Let T be a semiring such that either T is trivial or T is a parasemifield. Then S s T j  04 is a semifield of type 12.1Ž4. and every semifield of this type may be constructed in the described way. 13. PARASEMIFIELDS 13.1. PROPOSITION w41x. There exists a one-to-one correspondence between additi¨ ely idempotent parasemifields and lattice-ordered non-tri¨ ial abelian groups. Proof. If S is an ai-parasemifield, then S Ž⭈, n , k . is a lattice-ordered group, where a n b s a q b and a k b s Ž ay1 q by1 .y1 for all a, b g S. Conversely, if SŽ⭈, n , k . is a lattice-ordered group and a q b s a n b, then SŽq, ⭈ . is an ai-parasemifield. 13.2. Remark. Let S be an additively cancellative parasemifield and let Qq denote the sub-parasemifield of S generated by 1 S . Ži. Qq( ⺡q Žthe parasemifield of positive rationals.. Žii. Q s Qqj Qyj 04 is a subfield of the ring R s RŽ S . Žsee 4.2. and Q ( ⺡, so that R is an algebra over the field of rationals.

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Žiii. S is imbeddable into a field if and only if R is a domain and this is further equivalent to the condition that ab q 1 S / a q b for all a, b g S _ 1 S 4 . 13.3. EXAMPLE. Put S s ⺡q= ⺡q. Then S is an ac-parasemifield, and S is archimedean but S is not imbeddable into a field Žin particular, S is not conical.. 13.4. EXAMPLE. Let S be the set of fractions of the form frg, where f, g g ⺡qw x x. Then S is a subsemiring of ⺡Ž x . Žand hence S is imbeddable into a field., S is an ac-parasemifield but S is not conical Že.g., 1rx y 1 f RŽ S . : ⺡Ž x ... 13.5. PROPOSITION. Let S be a parasemifield. Define a relation ␩S on S by Ž a, b . g ␩S if and only if there exist a non-negati¨ e integer m and elements u, ¨ g S j  04 such that 2 m a s b q u and 2 m b s a q ¨ . Then Ži. ␩S is a congruence of S. Žii. Sr␩S is either tri¨ ial Ž and then S is additi¨ ely cancellati¨ e . or an ai-parasemifield Ž see 13.1.. Žiii. If a, b, c belong to a block of ␩S and b / c, then a q b / a q c. Živ. P s  a g S; Ž a, 1 S . g ␩S 4 is a subsemiring of S, P Ž⭈. is a subgroup of SŽ⭈., and either P is tri¨ ial Ž and then S is additi¨ ely idempotent . or P is an ac-parasemifield Ž see 13.2.. Proof. ŽIii. Let a q b s a q c. We have a q u s 2 m b, b q 2 m b s b q a q u s c q a q u s c q 2 m b, 2Ž b q 2 my 1 b . s b q b q 2 m b s b q c q 2 m b s c q c q 2 m b s 2Ž c q 2 my 1 b ., b q 2 my 1 b s c q 2 my1 b, . . . , 2 b s b q c. Quite similarly, 2 c s b q c, and hence b s c. The rest is clear.

14. FINITE AND FINITELY GENERATED SIMPLE SEMIRINGS 14.1. THEOREM. The following conditions are equi¨ alent for a semiring S: Ži. Žii. Žiii.

S is finite and congruence-simple. S is finite and ideal-simple.

S is Ž isomorphic to . one of the following semirings: Žiii1. the two-element semirings Z1 , Z2 , Z3 , Z4 , and Z5 ; Žiii2. finite fields; Žiii3.

zero-multiplication rings of finite prime order;

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Žiii4. the semirings Ž semifields. V Ž G . Ž see 3.2., G being a finite abelian group. Proof. Combine 3.1, 3.2, 3.3, 3.4, 5.3, and 11.2. 14.2. THEOREM. The following conditions are equi¨ alent for a semiring S: Ži. S is finitely generated, congruence-simple, and infinite. Žii. S is Ž isomorphic to . one of the following semirings: Žii1. the semirings Ž semifields. V Ž G . Ž see 3.2., G being an infinite finitely generated abelian group; Žii2. the semirings W Ž A. Ž see 5.1., A being a finitely generated subsemigroup of ⺢Žq. such that A l ⺢q/ ⭋ / A l ⺢y. Proof. Combining 3.1, 3.2, 3.3, 3.4, 4.2, 4.3, 5.1, and 5.3, we can restrict ourselves to the case when S is a finitely generated subsemiring of a field F such that F s  a y b; a, b g S4 . Then, of course, F is finitely generated as a ring and consequently F is finite Žthis is a rather well-known result and the reader may try to prove it as an exercise.. 14.3. COROLLARY. Let S be a congruence-simple semiring such that S is finitely generated but not additi¨ ely idempotent. Then S is finite. 14.4. Remark Žcf. w25x.. Ži. Let S be a congruence-simple semiring such that < S < ) 2 / 0 . Then either S is a field or S ( V Ž G . for an abelian group G Ž< G < s < S <.. In both cases, S is ideal-simple. Žii. Ž⺡q. ᑾ is an Žideal-simple. parasemifield for any cardinal number ᑾ G 1. 14.5. PROPOSITION. ring S: Ži. Žii. order.

The following conditions are equi¨ alent for a semi-

S is finitely generated, ideal-simple, and additi¨ ely cancellati¨ e. S is either a finite-field or a finite zero multiplication ring of prime

Proof. First, let S be an ac-parasemifield. We claim that S is not finitely generated as a semiring. Assume the contrary, put R s RŽ S . and take a maximal ideal I of R. The relation r defined on S by Ž a, b . g r if and only if a y b g I is a congruence of S and T s Srr is again an ac-parasemifield. On the other hand, F s RŽT . ( RrI is a finitely generated ring and a field. Then F is finite, and T Žq. is a group, a contradiction with 0 f T. Now, let S be a finitely generated id-simple ac-semiring. Then S is not a parasemifield and, in view of 12.1 and 12.4, S is not a proper semifield

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either. Thus, according to 11.2, S is a Žfinite. field or a zero multiplication ring of prime order. 14.6. Remark Žcf. 14.2, 14.3, 14.5.. It seems to be an open problem whether every infinite finitely generated ideal-simple semiring is additively idempotent.

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