The Characterization of Congruences on Additive Inverse Semirings 1

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1 International Journal of Algebra, Vol. 7, 2013, no. 18, HIKARI Ltd, The Characterization of Congruences on Additive Inverse Semirings 1 Ziqing Wei and Wenting Xiong Public Teaching Department, Jiangxi University of Technology Jiangxi, Nanchang, , P. R. China @qq.com Copyright c 2013 Ziqing Wei and Wenting Xiong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we provide a set of independent axioms characterizing the kernel of a congruence on an additive inverse semiring and show how to reconstruct the congruence from its kernel. Next we show that the mapping ρ trρ is a complete homomorphism of the congruence lattice of S onto the lattice of normal congruences on E + (S), and that the congruence induced by it has all its classes complete modular lattices. Mathematics Subject Classification: 16Y60, 20M10 Keywords: congruence pair, kernel normal system, lattice, complete modular lattice 1 Introduction A semiring S is defined as an algebra system (S, +, ), such that (S, +) and (S, ) are semigroups connected by a(b + c) =ab + ac and (b + c)a = ba + ca for all a, b, c S. A semiring S is a skew ring if its additive reduct (S, +) is a group. A semiring S is an additive inverse semiring if (S, +) is an inverse semigroup. 1 The research is supported by the Natural Science Research Project of Jiangxi University of Technology(ZR12YB17); the NSF of Jiangxi Province( ); the SF of Education Department of Jiangxi Province(GJJ09459) and the SF of Jiangxi Normal University(1905).

2 858 Ziqing Wei and Wenting Xiong A nonempty subset A of a semiring S is called an ideal of S if a+b A, sa, as A for all a, b A, s S. If S is an inverse semiring, we denote a the unique additive inverse of a, E + (S) the set of all additive idempotents. Easily, we can prove if S is an additive inverse semiring, then E + (S) is an ideal of S. Let S be an additive inverse semiring, σ a congruence on E + (S). σ is normal, in the sense that (i) eσf ( a S) ( a + e + a)σ( a + f + a); (ii) eσf ( c S) ecσfc and ceσcf. Since the intersection of a non-empty family of congruences on a general semiring S is a congruence on S, we can induce that for any relation R on S there is an unique smallest congruence R # containing R. We now give a result analogous to semigroup which will give us a usable description of R #. Let S be a semiring. Define a semiring S 0 as follows: if (S, +) has a zero element, then let S 0 = S; otherwise, define S 0 = S {0} to be a new semiring by compensating the operations of S as follows: For any relation R on S we denote ( x S 0 )x +0=0+x = x, x0 =0x =0. R c = {(u + xay + v, u + xby + v) u, v, x, y S 0, (a, b) R} {(u + a + v, u + b + v) u, v S 0, (a, b) R} {(u + xa + v, u + xb + v) u, v, x S 0, (a, b) R} {(u + ay + v, u + by + v) u, v, y S 0, (a, b) R}. If R is a relation on a set X, then the minimum equivalence containing R is R e =[R R 1 1 X ] = [R R 1 1 X ] n. If R is a relation on a semiring S, then R # =(R c ) e.soif{ρ i } is a family of congruences on a semiring S, then i ρ i =( i ρ i ). In fact, i ρ i =(( i ρ i ) c (( i ρ i ) c ) 1 1 S ) =(( i ρ c i ) ( i(ρ c i ) 1 ) 1 S ) =( i ρ i ). Semirings with special additive reduct are studied [3], [8], [6],. The congruences and the congruence pairs on an inverse semigroup[5],[2] and orthorings[7] are characterized respectively. Other definitions and results about semigroups and semirings are referred to [2],[4],[5] and [1]. In this paper, we only considered the additive inverse semirings and we will generalize some results about congruences, congruence pairs on semirings and the lattice of congruences and give some characterizations and structure of additive inverse semirings. n=1

3 Characterization of congruences Characterizations and Structure If ρ is a congruence on an additive inverse semiring S, let K(ρ) ={eρ e E + (S)}. Such collections of subsets of S can be characterized abstractly, and we thus have the following concept. Definition 2.1. Let K be a family of pairwise disjoint additive inverse subsemirings of an additive inverse semiring S. Then K is a kernel normal system for S, ifk satisfies (i)e + (S) K K K. (ii)for each a S and K K, there exits L Ksuch that ( a)+k + a L. (iii)if a, a + b, b +( b) K, K K, then b K. (iv)if a L, b T, L, T K, there exists U Ksuch that ab U. (v)for e, f E + (S), if e, f L, L K, there exist K 1,K 2 K such that ec, fc K 1, ce, cf K 2 for each c S. For such a family K, we define a relation ξ K on S by aξ K b a +( a),b+( b),a+( b) K for some K K. With this notation and definition, we have one new characterization of congruences on additive inverse semiring as follows. Theorem 2.2. Let S be an additive inverse semiring. If K is a kernel normal system for S, then ξ K is the unique congruence ξ on S for which K(ξ) =K. Conversely, if ξ is a congruence on S, then K(ξ) is a kernel normal system for S and ξ K (ξ) =ξ. Proof. First, we shall prove the direct part of the theorem. Let K be a kernel normal system for S, let K = L K L and define τ on E + (S) by eτf e, f L for some L in K. We now verify that (K, τ) is a congruence pair for S and that ξ K = ρ (K,τ). Since the members of K are pairwise disjoint and 2.1(i) holds, we obtain that τ is an equivalent relation. Condition 2.1(ii) and (v) implies that τ is a normal congruence. We now show that K is closed under multiplication. Hence let a L and b T for L, T K. Let e =( a) +a and f = b +( b). Since τ is a normal congruence, there exists U K such that E + L + E+ T E+ U. But then e + E+ T + e E+ U, which by 2.1(ii) implies that e + T + e U. Nowe + b + e e + T + e U, which implies that (e + b + e)+(e +( b)+e) U. Also, (e + b + e)+(e +( b)+e)+(e + b) =e + b + e +( b)+e + b = e +(b + e +( b)) + e + b = e + b + e +( b)+b = e + b +(( b)+b)+e = e + b + e U,

4 860 Ziqing Wei and Wenting Xiong and (e + b) +(( b) +e) =e + b +( b) E + L + E+ T U. Now applying 2.1(iii), we get e + b U. Since E + T + E+ L = E+ L + E+ T E+ U and a L, for L is an additive inverse subsemiring of S, we can similarly prove that f +( a) U. Thus a + f = (f +( a)) U and we obtain a + b =(a + e)+(f + b) =(a + f)+(e + b) U + U U. and applying 2.1(iv), we get ab V for some V K. So K is a subsemiring of S. Since each L in K is closed under taking of additive inverses, so is K, and is thus an additive subsemiring of S. Condition 2.1(i)and (ii) insure that K is full and self-conjugate. Hence K is a normal subsemiring of S. We now verify (K, τ) is a congruence pair next. Hence let a S, e E + (S),a + e K and eτ( a)+a. Then a + e L and e, ( a)+a T for some L, T K. It follows that e +( a) = (a + e) L and thus e +( a)+a + e = e +(( a)+a)+e L T, so that L = T. Hence e, e +( a), ( a)+a L, which by 2.1(iii) gives a L, so that also a L, then a K, as required. If a K, then a L for some L K, and thus a +( a), ( a)+a L and hence a +( a)τ( a)+a. Consequently, (K, τ) is a congruence pair. We now check that ρ (K,τ) = ξ K. First let aρ (K,τ) b, so that ( a)+aτ( b)+b and a +( b) K. Hence ( a)+a, ( b)+b L and a +( b) T for some L, T K. By 2.1(ii), we have a + L +( a) L 1 and b + L +( b) L 2 for some L 1,L 2 K. Now a +( b)+b +( a) =a +(( b)+b)+( a) T L 1 so that T = L 1 ; also b +( a)+a +( b) =b +(( a)+a)+( b) T L 2 and hence T = L 2. Since also a +( a) =a +(( a)+a)+( a) L 1 and similarly b +( b) L 2, we deduce that a +( a),b+( b),a +( b) T and thus aξ K b. Conversely, let aξ K b. Then a +( a),b+( b),a+( b) L for some L K. We have ( a)+l + a L 1 and ( b)+l + b L 2 for some L 1,L 2 K.Now ( a)+a +( b)+b =( a)+[a +( b)+(b +( a))] + a =( b)+[b +( a)+(a +( b))] + b L 1 L 2 since a +( b),b+( a) L. It follows that L 1 = L 2. Since ( a)+a =( a)+(a + ( a)) + a L 1 and similarly ( b) +b L 2, we deduce that ( a) +a, ( b) +b L 1 = L 2. We thus have ( a)+aτ( b)+b and a +( b) K which gives aρ (K,τ) b. We verify next that K(ξ K )=K. First let L K. Then L is an additive inverse subsemiring of S. In order to show that L K(ξ K ), it suffices to show that L is a ξ K class. Ifa, b L, then a+( a),b+( b),a+( b) L and thus aξ K b. Let a L and aξ K b. Then a+( a),b+( b),a+( b) T for some T K. But a+( a) L T, so that L = T. Since aξ K b, we also have ( a)+a, ( b)+b, ( a)+b U for

5 Characterization of congruences 861 some U K, and as above, U = L. Thus, in particular, ( a), ( a)+b, b +( b) L and 2.1(iii) implies that b L. Consequently, L is a ξ K -class and thus L K(ξ K ). Conversely, let L K(ξ K ). Then L contains an idempotent e, and thus e T for some T Kby 2.1(i). By the above, T also is a ξ K -class, and we must have L = T. Consequently, L K, which completes the verification that K(ξ K )=K. If now ρ is any congruence on S for which K(ρ) =K, then we have ker ρ =kerξ K and trρ = trξ K, and we get ρ = ξ K. This establishes the uniqueness of ξ K and complete the proof of the direct part of the theorem. Second, we shall prove the converse part. Indeed, let ξ be a congruence on S. It is obvious that K(ξ) consists of a family of pairwise disjoint additive inverse subsemiring of S whose union contains E + (S). Let a S and let K be a ξ-class containing an idempotent e. Then for any K K, ( a)+k +aξ( a)+e+a so that ( a)+k + a L where L is the ξ-class containing the idempotent ( a)+e + a. This verifies 2.1(ii). With the same notation, assume a, a + b, b +( b) K. Then b =(b +( b)) + bξe + bξa + bξe so that b K. This verifies 2.1(iii). Since ξ is a congruence, condition 2.1(ii) and (v) can be easily proved. Hence we complete the proof that K(ξ) is a kernel normal system. By part of the proof above, we have K(ξ K (ξ)) = K(ξ). Thus ξ K (ξ) and ξ have the same kernel normal system, so by the uniqueness proved in the theorem, we obtain ξ K (ξ) =ξ. By the introduction about the description of R #, we can get: Proposition 2.3. [7] Let ρ, σ be congruences on a semiring S. Then ρ σ =(ρ σ). For any additive inverse semiring S, let C(S) be the lattice of all congruences on S and let N(E + (S)) be the lattice of all normal congruences on E + (S). Corollary 2.4. Let S be an additive inverse semiring. Then the join in C(E + (S)) of two normal congruences on E + (S) is a normal congruence on E + (S). Proof. Let ρ, σ C(E + (S)). Applying proposition 2.3, we get ρ σ =(ρ σ). Let e, f E + (S), then for any a S and c S, e(ρ σ)f e(ρ σ)a 1,a 1 (ρ σ)a 2,..., a n 1(ρ σ)f. a + e + a(ρ σ) a + a 1 + a,..., a + a n 1 + a(ρ σ) a + f + a and ec(ρ σ)a 1 c,..., a n 1 c(ρ σ)fc,ce(ρ σ)ca 1,..., ca n 1 (ρ σ)cf. ( a)+e + a(ρ σ)( a)+f + a and ec(ρ σ)fc,ce(ρ σ)cf. Thus ρ σ is a normal congruence on E + (S). Remark 2.5. Let S be an additive inverse semiring. For any congruence ρ on S, define a relation ρ min on S by aρ min b a + e = b + efor some e E + (S),eρ( a)+aρ( b)+b.

6 862 Ziqing Wei and Wenting Xiong Similarly to the proof in semigroup, we now give a result in semiring: Proposition 2.6. Let K be a sublattice of the lattice (C(S),,, ) of congruences of a semiring S, and suppose that ρ σ = σ ρ for all ρ, σ in K. Then K is a modular lattice. We are now ready for the result concerning the lattice of congruences in relation to their traces. Theorem 2.7. Let S be an additive inverse semiring. Define a mapping tr by tr : ρ trρ (ρ C(S)). Then tr is a complete homomorphism of C(S) onto N(E + (S)). Letθ be the congruence on C(S) induced by tr. Then for any ρ C(S), ρ min is the least element of ρθ and ρθ is a complete modular sublattice of C(S). Proof. Let F be a nonempty family of congruences on S. Then for any e, f E + (S), we obtain etr( ρ F ρ)f e( ρ F ρ)f eρf for all ρ F e trρ f for all ρ F e( ρ F trρ)f, which proves tr( ρ F ρ)= ρ F trρ. Using proposition 2.4, we get etr( ρ F ρ)f e( ρ F ρ)f eρ 1 x 1,x 1 ρ 2 x 2,..., x n 1 ρ n f for some x i S, ρ i F eρ 1 x 1 +( x 1 ),x 1 +( x 1 )ρ 2 x 2 +( x 2 ),..., x n 1 +( x n 1 )ρ n f e( ρ F trρ)f, which shows that tr( ρ F ρ) ρ F trρ. The converse inclusion follows easily: e ρ F trρf etrρ 1 e 1,e 1 trρ 2 e 2,..., e n 1 trρ n f eρ 1 e 1,e 1 ρ 2 e 2,..., e n 1 ρ n f e( ρ F ρ)f etr( ρ F ρ)f. This proves tr( ρ F ρ)= ρ F trρ. Consequently, tr is a complete homomorphism of C(S) inton(e + (S)). Let τ N(E + (S)) and define ρ on S by aρb a + e = b + e for some e E + (S),eτ( a)+aτ( b)+b. It is obvious that ρ is an equivalence relation on S. Let aρb and c S. Then (c+a)+c+a =( a)+(( c)+c)+a =(( a)+a)+[( a)+(( c)+c)+a]τe+( a)+(( c)+c)+a, (c + b)+c + b =( b)+(( c)+c)+b

7 Characterization of congruences 863 =(( b)+ b)+( b)+(( c)+ c)+b +(( b)+b)τe +( b)+(( c)+c)+b + e = e +( a)+(( c)+c)+a + e = e +( a)+(( c)+c)+a, which shows that e+( a)+(( c)+c)+aτ (c+a)+c+aτ (c+b)+c+b for e+( a)+(( c)+c)+a E + (S). Also c + a + e +( a)+(( c)+c)+a = c + b + e +( a)+(( c)+c)+a. So we obtain c + aρc + a. Further, (a + c)+a + c =( c)+(( a)+a)+cτ( c)+e + c, (b + c)+b + c =( c)+(( b)+b)+cτ( c)+e + c, which shows that ( c)+e + cτ (a + c)+a + cτ (b + c)+b + cfor( c)+e + c E + (S). Also a + c +( c)+e + c = b + c +( c)+e + c, and thus a + cρb + c. Also we have ac + ec =(a + e)c =(b + e)c = bc + ec for ec E + (S), ecτ(( a)+a)cτ(( b)+b)c, ca + ce = c(a + e) =c(b + e) =cb + ce for ce E + (S), ceτc(( a)+a)τc(( b)+b), where we have used normality of τ. So that caρcb, cbρca. It follows that ρ is a congruence on S. Further, for any e, f E + (S), eρf e + gτe + f for all g E + (S) eτf. so that trρ = τ which also shows that tr maps C(S) onton(e + (S)). It is easy to verify that ρ min is an equivalent relation. Let a+e = b+e for some e E + (S),eρ( a)+aρ( b)+b and c S. Then (c+a)+c+a =( a)+(( c)+c)+a =(( a)+a)+[( a)+(( c)+c)+a]ρe+( a)+(( c)+c)+a, (c+b)+c+b =( b)+(( c)+c)+b =(( b)+b)+( b)+(( c)+c)+b+(( b)+b) ρe+( b)+(( c)+c)+b + e = e +( a)+(( c)+c)+a + e = e +( a)+(( c)+c)+a, which shows that e+( a)+(( c)+c)+aρ (c+a)+c+aρ (c+b)+c+b for e+( a)+(( c)+c)+a E + (S).

8 864 Ziqing Wei and Wenting Xiong Also c + a + e +( a)+(( c)+c)+a = c + b + e +( a)+(( c)+c)+a. So we obtain c + aρ min c + b. Further, which shows that (a + c)+a + c =( c)+(( a)+a)+cρ( c)+e + c, (b + c)+b + c =( c)+(( b)+b)+cρ( c)+e + c, ( c)+e + cρ (a + c)+a + cρ (b + c)+b + cfor( c)+e + c E + (S). Also a + c +( c)+e + c = b + c +( c)+e + c, and thus a + cρ min b + c. Also we have ac + ec =(a + e)c =(b + e)c = bc + ec for ec E + (S), ecρ(( a)+a)cρ(( b)+b)c, ca + ce = c(a + e) =c(b + e) =cb + ce for ce E + (S), ceρc(( a)+a)ρc(( b)+b). So that caρ min cb, cbρ min ca. Consequently, ρ min is a congruence on S. Ifa+e = b+e and eρ( a) +aρ( b) +b, then aρa + e = b + eρb. This proves that ρ min ρ. If e, f E + (S) and eρf, then e +(e + f) =f +(e + f) and e + fρeρf so that eρ min f. This shows that trρ = trρ min, and since the definition of ρ min depends only on idempotents, it follows that ρ min is the least element of ρθ. Next we show that ρθ is the complete sublattice of S. For any ρ i ρθ, we have trρ i = trρ. Then tr( ρi ρθ)ρ i = ρi ρθtrρ = trρ = trρ. Similarly, tr( ρi ρθ)ρ i = trρ, hence ρ i, ρ i ρθ. So that ρθ is the complete sublattice of S. Now let ρ, λ C(S) be such that ρθ = λθ, and let aρλb. Then aρc and cλb for some c S. Hence a +( a)ρc +( c) and thus a +( a)λc +( c) since trρ = trλ. Also c +( c)λb +( c) and thus a +( a)λb +( c) which yields aλb +( c)+a. A similar argument shows that bρb +( c)+a and thus aλρb. Hence ρλ λρ and by symmetry, we deduce that ρλ = λρ. An application of proposition 2.8 gives that ρθ is a modular lattice. References [1] J. M. Howie, Fundamentals of semigroup theory, Oxford: Clarendon Press, [2] M. Patrich, Inverse semigroups, Semigroup Forum, Vol. 32(1985), [3] P.H. Karvellas, Inverse semirings, J. Austral Math. Soc. 18(1974),

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