On Epimorphically Closed Homotypical Permutative Varieties

We find some sufficient conditions that a homotypical identity be preserved under epis of semigroups in conjunction with any nontrivial permutation identity. INTRODUCTION The general question of which identities are preserved under epis has been studied in semigroup theory, ring theory and elsewhere (Burgess, 1975). For example, in (Gardner, 1979) Gardner has shown that certain identities weaker than commutativity are not preserved under epis of rings although commutativity is preserved under epis of rings (Bulazewska, 1965). It has been shown by Higgins (Higgins, 1984), that identities for which both sides contain repeated variables are not preserved under epis of semigroups. In (Howie, 1967), Howie and Isbell have shown that commutativity is preserved under epis of semigroups. The author (Khan, 1982, Khan, 1985) has extended this result and shown that all identities in conjunction either with commutativity or semicommutative permutation identity are preserved under epis of semigroups. In (Clfford, 1967, Proof of Theorem 8.3), Higgins has shown that, in general, all homotypical identities for which both sides contain repeated variables are not preserved under epis in conjunction with any non trivial permutation identity. In (Khan, 1985, Theorem 4.7), the author has found some sufficient conditions that a homotypical identity containing repeated variables on both sides be preserved under epis of semigroups in conjunction with any nontrivial permutation identity. In this paper we extend further [Khan, 1985, Theorem 4.7(iii)] by finding some sufficient conditions that a homotypical identity containing repeated variables on both sides be preserved under epis of semigroups in conjunction with any nontrivial permutation identity. However, finding a complete determination of all identities containing repeated variables on both sides which are preserved under epis of semigroups still remains an open problem. Preliminaries: $PRUSKLVPV $ % LQ WKH FDWHJRU\ of semigroups is called an epimorphisms (epi for short) if ∀ C ∈ C and for all morphisms : B C, implies = . It can be easily verified that a morphism S T is epi if and only if the inclusion map i: S T is epi, and the inclusion map i: U S from any subsemigroup U of S is epi if and only if Dom (U, S) = S. KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY VOL.I, No.III, JANUARY, 2007. 2 The following, Isbell’s zigzag theorem, provides a most useful characterization of semigroup dominions. Result 2.1([9,Theorem 2.3] or [7,Theorem VII.2.13]). Let U be a subsemigroup of any semigroup S and let d be any element of S. Then d ∈ Dom (U, S) if and only if either G U or there are elements a0, a1, a2, ...,a2m ∈ U; t1, t2, tm, y1, y2,..., ym ∈ S such that d = a0 t1, a0 = y1a1 a2i-1ti = a2iti+1, yia2i = yi+1a2i+1 (i = 1, 2,3, ...,m1) (1) a2m-1tm = a2m, yma2m = d. These equations are called a zigzag of length m over U with value d and spine a0, a1, a2, ...,a2m.. An identity of the form x1x2x3...xn = xi1xi2...xin (n• is called a permutation identity, where i is any permutation of the set {1,2,...,n}. Again a permutation identity of the form x1x2x3...xn = xi1xi2 ...xin (2) is called nontrivial if i is any nontrivial permutation of the set {1,2,...,n}. Further a permutation identity is said to be semicommutative if i1  DQG in  n. A semigroup S is said to be permutative if it satisfies a nontrivial permutation identity (2) and a permutative semigroup S is said to be semicommutative if i1  DQG in  n. An identity u = v is said to be preserved under epis if for all semigroups U and S with U a subsemigroup of S and such that Dom(U, S) = S, U satisfying u = v implies S satisfies u = v. Result 2.2 ([12],Theorem 3.1) All permutation identities are preserved under epis. Result 2.3 ([11], Result 3). Let U be any subsemigroup of a semigroup S. Then for any G Dom(U, S)\U, if (1) be a zigzag of shortest possible length m over U with value d, then yj, tj ∈ S\U ∀ j = 1,2,...,m. In the following results, let U and S be any semigroups with U a subsemigroup of S and such that Dom(U, S) = S. KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY VOL.I, No.III, JANUARY, 2007. 3 Result 2.4 ([11], Result 4). For any d ∈ S\U, if (1) be a zigzag of shortest possible length m over U with value d and k be any positive integer, then there exist a1, a2, ..., ak ∈ U and dk ∈ S\U such that d = a1 a2... ak dk. Result 2.5 ([11],Corollary 4.2). If U be permutative, then s x1 x2...xkt = sxj1xj2 ...xjkt ∀ x1, x2, ...xk ∈ S, s, t ∈ S\U, and any permutation j of the set {1,2,...,k}. Result 2.6 ([12], Proposition 4.6). Let U be a permutative semigroup. If d ∈ S\U and (1) be a zigzag of length m over U with value d and with y1 ∈ S\U(for example if the zigzag (1) is of the shortest possible length), then dk = a0 t1 k for any positive integer k. The notations and conventions of Cllifford and Preston or Howie will be used throughout without explicit mention. 3. Main Result: An identity u = v is said to be homotypical if C(u) = C(v); where C(u), for any word u, is the set of all variables appearing in u; otherwise heterotypical. Theorem 3.1. Let (2) be any nontrivial permutation identity. Then any nontrivial homotypical identity I (one which is not satisfied by the class of all semigroups) of the following form is preserved under epis in conjunction with (2): x1 p 1x2 p 2...xr p r=x1 q 1x2 q 2...xr q r, (*) where 0 < pr ” pr-1 ”... ” p2 ” p1 and 0 < q1 ” q2”«” qr-1” qr and r • 0. Proof: Take any semigroups U and S with U epimorphically embedded in S, and such that U(and hence, S, by Result 2.2) satisfies the identity (2). We show that the identity(*) satisfied by U is also satisfied by S. Assume that U satisfies the given identity(*). For k = 1,2,...,r; consider the word x1 p 1x2 p 2 ...xk p k of length p1 + p2 + ... + pk. We shall prove that S satisfies (*) by induction on k , assumming that the remaining elements x k+1, x k+2,..., xr ∈ U. First for k = 0, the identity is satisfied by S vacuously. So assume next that the identity(*) is satisfied ∀ x1, x2,..., xk-1 ∈ S and ∀ xk, xk+1,..., xr ∈ U. Without loss we can assume that xk ∈ S\U. As xk ∈ S\U and Dom(U, S) = S, by Result 2.1, we may let (1) be a zigzag of shortest possible length m over U with value xk. We assume first that 1 < k < r. Now x1 p 1x2 p 2...xr p r = x1 p 1x2 p 2 ... xk-1 p k-1a0 p kt1 p kxk+1 p k+1... xr p r (by eq. (1) and Result 2.6) KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY VOL.I, No.III, JANUARY, 2007. 4 = x1 p 1x2 p 2 ... xk-1 p k-1a0 p kbk+1 (1)p kbk+2 (1)p k...br (1)p kt1 (1)p kxk+1 p k+1...xr p r (by eq. (1) and Results 2.4 and 2.5, for some bk+1 ,bk+2 , br (1) ∈ U and t1 ∈ S\U) = x1 p 1x2 p 2 ... xk-1 p k-1a0 p kbk+1 (1)p k+1bk+2 (1)p k+2...br (1)p rw t1 (1)p kz (where w = bk+1 (1){p k -p k+1 bk+2 (1){p k -p k+2 ...br (1){p k -p r } and z = xk+1 p k+1...xr p r, by Result 2.5) = x1 q 1x2 q 2 ...xk-1 q k-1a0 q kbk+1 (1)q k+1bk+2 (1)q k+2...br (1)q rw t1 (1)p kz (by inductive hypothesis as a0 ∈ U) = vy1 (1)q k c1 (1)q kc2 (1)q k ...ck-1 (1)q ka1 q kbk+1 (1)q k+1bk+2 (1)q k+2...br (1)q rw t1 (1)p kz (by eq.(1) and Result 2.5 and dual of Result 2.4, for some c1 ,c2 ,...,ck-1 (1) ∈ U; where v = x11x22 ...xk-1 q k-1 and y1 , t1 (1) ∈ S\U) = vy1 (1)q kv c1 (1)q 1c2 (1)q 2...ck-1 a1 bk+1 (1)q k+1bk+2 (1)q k+2...br (1)q rw t1 (1)p kz (by Result 2.5, where v = c1 (1){q k -q 1 } c2 (1){q k -q 2 } ...ck-1 (1){q k -q k-1 } as y1 , t1 (1) ∈ S\U =y1 (1)q kv c1 (1)p 1c2 (1)p 2...ck-1 (1)p k-1a1 p kbk+1 (1)p k+1bk+2 (1)p k+2...br (1)p rw t1 (1)p kz (as U satisfies (*)) = vy1 (1)q kv c1 (1)p 1c2 (1)p 2...ck-1 (1)p k-1a1 p kbk+1 (1)p kbk+2 (1)p k...br (1)p kw t1 (1)p kz (by Result 2.5, as y1 , t1 (1) ∈ S\U and w = bk+1kk+1 bk+2kk+2 ...brkr ) = vy1 (1)q kv c1 (1)p 1c2 (1)p 2...ck-1 (1)p k-1a1 p k t1 (1)p kz (by Result 2.5, as t1 (1)p k = bk+1 (1)p kbk+2 (1)p k ...br (1)p kt1 (1)p k) = vy1 (1)q kv c1 (1)p 1c2 (1)p 2...ck-1 (1)p k-1a2 p k t2 p kz (by Result 2.5 and equations (1))