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Indian Journal of Mathematics and Mathematical Sciences Vol. 7, No. 1, (June 2011) : 39-43
CONGRUENCE ON SEMIGROUPS G. Shobhalatha & A. Rajeswari bhat
ABSTRACT This paper contains some results on congruence semigroups and -reflexive semigroups using the properties like weakly separative , weakly commutative, quasi commutative etc. Defining a relation ‘’ on a commutative semigroup ‘S’ by ab => abn = bn + 1, ban = an + 1 for some positive integer ‘n’ and any a, b in S, it is proved that ‘’is a separative congruence on ‘S’. Also it is proved that if ‘S’ is quasi commutative semigroup, and ‘’ be the congruence relation on ‘S’ then S/ is a maximal separative homomorphic image of S. On the other hand using the permutable property it is proved that a -reflexive semigroup is cancellative with or without considering separative condition . This paper also contains some results on weakly commutative semigroups. One of the properties like permutable, separative etc has taken over a weakly commutative semigroup and proved that it is weakly cancellative or cancellative.
1. INTRODUCTION The concept of congruence semigroups is introduced by Pondelicek [1]. It is proved that if ‘S’ is a left weakly commutative semigroup and ‘σ’ is a congruence relation on ‘S’ then ‘σ’is separative congruence on ‘S’, also if ‘S’ is a duo semigroup then S/π is a maximal separative homomorphic image of S. σ-reflexive semigroup is introduced by Chacron and Thierren (1972) [2]. It is proved that any semigroup ‘S’ is σ-reflexive if it satisfies the condition (ab) = (ba)m for some positive integer m > 1. The research of weakly commutative semigroups was being began from famous paper Pondelicek [1]. In commutative semigroups, Pondelicek [1] extend his results on quasi commutative semigroups to weakly commutative and duo semigroups where in particular they proved that every commutative semigroup is weakly commutative. The research of separative semigroups was being began from the famous paper of Howitt [4] and Zuclevman [4]. They proved that any commutative separative semigroup is isomorphic to a semilattice of cancellative semigroups. We extended his results on weakly commutative semigroups to weakly cancellative and separative semigroups. 2. PRELIMINARIES In this section some basic concepts of semigroups and other definitions needed for the study of congruence semigroups are presented.
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G. Shobhalatha & A. Rajeswari bhat
Definition 2.1: A semigroup (S, .) is said to be separative if x2 = xy, y2 = yx => x = y and x2 = yx, y2 = xy => x = y for all x, y in S. Theorem 2.2: Let (S, .) be a commutative semigroup. Define a relation ‘σ’ on a semigroup S by a σ b => abn = bn + 1, ban = an + 1, for any positive integer n and for any a, b € S, then ‘σ’ is a separative congruence on S. Proof: First we prove that ‘σ’ is an equalance relation on S. We know that a .an = a = ana for all a, b in S => a σ a. ‘σ’ is reflexive. Let a σ b for some a, b in S then abn = bn + 1, ban = an + 1. Replace a by b and b by a we have ban = an + 1, abn = bn + 1 => b σ a. ‘σ’ is symmetric. Now we prove that it is transitive. Suppose a σ b and b σ c. This implies abn = bn + 1 , ban = an + 1 — (1) also bcm = cm + 1 cbm = bm + 1 — (2). From second equation b icm = cm + i for any positive integer ‘i’, so that abn cm = bn + 1 cm = cm + n + 1 => acm + n = cm+n+1. Similarly cam + n = am + n + 1 from equation (1) => a σ c. σ is transitive. There fore σ is equalance relation. Now we prove that σ is congruence on S. Let a σ b => abn = bn + 1, ban = an + 1. To prove that ac σ bc, ca σ cb. Consider (ac) (bc)n = acbncn = cabncn = cbn + 1cn = bn + 1cn + 1 = (bc)n + 1 => ac σ bc. n+1
Similarly (ca)(cb)n = cacnbn = cabncn = cbn + 1 cn = cn + 1bn + 1 = (cb)n + 1 => caσcb=> s is congruence on S. Let a2σ ab σ ba σ b2 for any a, b in S. It follows that ab σ b2 => (ab) (b2)m = (b2)m + 1 for positive integer m, so ab2m + 1 = b2m + 2. Since s is congruence we have ba σ b2 => ba2 σ b2a σ (ba) b σ (ab) b σ a2b σ a (ab) s aa2 σ a3. There fore ba2 σ a3 => ba2(a3)k = (a3)k + 1 => ba3k + 2 = a3k + 3. Therefore ban = an + 1Similarly we prove that abn = bn + 1 => a σ b. σ is separative congruence on S. Definition 2.3: A semigroup (S, .) is said to be weakly commutative if for any a, b in ‘S’ we have (ab)k = xa = by for some x, y in ‘S’and a positive integer k. Theorem 2.4: Let (S, .) be a weakly commutative permutable semigroup. Define a relation σ on S as in the above theorem then σ is separative congruence on S. Proof: Given that (S, .) is weakly commutative. Then for any a, b, c in S and k is a positive integer (ab)k = ca = bc. To prove that σ is separative congruence on ‘S’. We know that σ is equalance relation on S (theorem 2.2) Now we prove that σ is congruence on S. Let a σ b => abn = bn + 1, ban = an + 1. To prove that ac ó bc, ca σ cb. Let (ac) (bc)m = ca (bc)m = bc (bc)m = (bc)m + 1 => ac σ bc. Similarly (ca) (cb)m = bc (cb)m = bc cmbm = bcm + 1bm = cm + 1b1 + m = (cb)m + 1 => ca σ cb. σ is congruence on S. To prove that σ is separative congruence let a2σ ab σ ba σ b2 for any a, b in S. It follows that ab σ b2 => (ab) (b2)m = (b2)m + 1 for positive integer m, so ab2m+1= b2m+2. Since s is congruence we have ba3 = (ba) a2 = (ab) a2 σ b2a2 σ a2a2 => ba3 σ a4 => ba3(a4)k = (a4)k + 1 => ba4k + 3 = a4k + 4. Therefore ban = an + 1. Similarly we prove that abn = bn + 1 => a σ b. σ is separative congruence on S. Definition 2.5: A semigroup S is called quasi commutative if for any a, b in S we have ab = bra for some positive integer r.
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Theorem 2.6: Let (S, .) be a quasi commutative semigroup then S/σ is a maximal separative homomorphic image of S. Proof: Suppose φ be an arbitary separative congruence on a quasi commutative semigroup we first prove that, let a, b in S if abn φ bn + 1, ban φ an + 1 for a positive integer n then a φ b, for n = 1, ab φ b2, ba φ a2 then a φ b. Assume that the results holds for n ≥ 1, abn + 1 φ bn + 2, ban + 1 φ an + 2. Now (abn)2 = (abn) (abn) = a (bna) bn = a (ab) bn = aabn + 1 φ abn + 2 φ abbn + 1 φ bnabn + 1 φ bnbn + 2 φ b2n + 2 φ (bn + 1)2 => abn φ bn + 1. Simlarly we prove ban φ an + 1 By induction a φ b. Since σ is congruence relation on S we have a σ b => abn = bn + 1, ban = an + 1 for a positive integer n => abn φ bn + 1, ban φ an + 1 => a φ b => σ ⊆ φ. By homomorphic theorem it is easy to see that s/φ is a maximal homomorphic image of S. Theorem 2.7: If (S, .) be a semigroup and let ‘φ’ be any congruence relation on S then φ is a maximal separative congruence S. Proof: Let φ be an arbitary separative congruence on a semigroup we first prove the following Lemma. Let a, b in S if abn φ bn + 1, ban φ an + 1 for a positive integer n then a φb for n = 1, ab φ b2, ba φa2 then a φ b. Assume that the results holds for n >= 1, abn + 1 φ bn + 2, ban + 1 φ an + 2. Now (abn)2 = (abn) (abn) φ bn + 1bn + 1 φ (bn + 1)2 => abn φ bn + 1. Simlarly we prove ban φ an + 1. By induction a φ b. Let φ be an arbitrary separative congruence on a semigroup S, if a φ b => abn = bn + 1, ban = an + 1 for a positive integer n and so abn φ bn + 1, ban φ an + 1 then a φ b => σ ⊆ φ, Hence σ is a maximal separative congruence on S. 3. STRUCTURE OF -REFLEXIVE AND WEAKLY SEPARATIVE SEMIGROUPS Definition 3.1: A semigroup (S, .) is said to be weakly separative if x2 = xy = y2 => x = y for all x, y in S. Theorem 3.2: Let (S, .) is σ-reflexive, weakly separative and permutable semigroup then (S, .) is cancellative. Proof: Given that (S, .) is σ-reflexive for every a, x, y in S, xa = (ax)m and ya = (ay)n for some positive integer m, n. Also (S, .) is permutable, weakly separative. To prove that (S, .) is cancellative that is xa = ya => x = y. Let xa = ya. Consider xa = (ax)m = am xm => xm = xa1 – m and ya = (ay)n = an yn => yn = ya1 – n. Consider xm + 1 = xm x = xa1 – mx = (xa) a– m x = (ya) a– m x = ya1 – m x = yxa1 – m = xm y. y n + 1 = y n y = ya1 – n y = (ya) a– ny = (xa) a – n y = xa1 – n y = xya1 – n = yn x. Now for m > 1x2m – 2 xy = xm – 2 xm + 1y = xm – 2 xm yy = x2m – 2 y2 = (xm – 1y)2x2m – 2xy = xm – 1 xmy = xm – 1xm + 1 = (xm)2. So (xm – 1 y)2 = (xm)2 => xm – 1 y = xm. For m = 2, x2 = xy. Similarly we prove that y2 = xy => x2 = xy = y2 => x = y. There fore (S, .) is cancellative.
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Definition 3.3: A semigroup (S, .) is said to be permutable if for every a, b, c in S, abc = acb = bac. Theorem 3.4:Let (S, .) is σ-reflexive and permutable semigroup then (S, .) is cancellative Proof: Given that (S, .) is σ-reflexive for every a, x, y in S, xa = (ax)m and ya = (ay)n for some positive integer m, n. Given that (S, .) is permutable. To prove that (S, .) is cancellative that is xa = ya => x = y. Let xa = ya. Consider xa = (ax)m = am xm => xm = xa1– m. Given ya = (ay)n = an yn => yn = ya1 – n. Consider xm + 1 = xm x = xa1 – m x = (x a) a– m = (ya) a– m x = ya1 – m x = yxa1 – m = xm y . yn + 1 = yn y = ya1 – n y = (ya) a– n y = (xa) a– n y = xa1 – n y = xya1 – n = yn x.Now for m > 1, x2m – 2 x2 = xm – 2 xm x2 = xm – 2 xa1 – mx2 = xm – 2(xa) a– m x2 = xm – 2 (ya) a– m x2 = xm – 2ya1 – m xx = xm – 2 yxa1 – m x = xm – 2 yxm x = xm – 2 yxm + 1 = xm – 2 yxm y = x2m – 2 y2 = (xm – 1 y)2. x2m – 2 x2 = xm – 2 xm x2= xm – 2 xm + 2 = (xm)2. So (xm – 1 y)2 = (xm)2 => xm – 1 y = xm. For m = 2 . x2 = xy. Similarly we prove that y2 = xy => x2 = xy = y2=> x2 = y2=> x = y. (S, .) is cancellative. 4. PROPERTIES OF WEAKLY COMMUTATIVE SEMIGROUPS Definition 4.1: A semigroup (S, .) is said to be weakly commutative if for any a, b in S we have (ab)k = xa = by for some x, y in S and a positive integer k. Definition 4.2: A semigroup (S, .) is said to be weakly cancellative if for every a, b, x, y in S ax = ay and xb = yb implies x = y. Theorem 4.3: If (S, .) is weakly commutative and permutable then (S, .) is weakly cancellative. Proof: Given that (S, .) is weakly commutative and permutable. To prove that (S, .) is weakly cancellative. Let xa = xb, ay = by. Consider (ab)m = xa => am bm = xa => bm = xa1 – m (ab)m = by => am bm = by => am = b1 – m y. Consider bm + 1 = bm b = x a1 – m b = xba1 – m = xaa1 – m = xa1 – m a = bm a . am+1= am a = b1 – m ya = b1 – m ay = b1 – m by = b1 – m yb = am b. Now for m > 1, b2m – 2 b2 = bm – 2 bm b2 = bm – 2 xa1 – m b2 = bm – 2 a1 – m xbb = bm – 2 a1 – m xab = bm – 2 a1 – m xba = bm – 2 a1 – m xaa = bm – 2 xa1 – m a2 = bm – 2 a2 bm = a2 b2m – 2 = (abm – 1)2. b2m – 2 b2 = bm – 2 bm b2 = bm – 2 bm + 2 = (bm )2. So (abm – 1)2 = (bm )2 => abm – 1 = bm. For m = 2 , b2 = ab. Similarly we prove that a2 = ab => a2 = ab = b2 => b2 = a2, a = b. (S, .) is cancellative. Theorem 4.4: Let (S, .) is weakly commutative and permutable and separative then (S, .) is weakly cancellative. Proof: Given that (S, .) is weakly commutative, permutable and separative. To prove that (S, .) is weakly cancellative. Let (ab)m = xa => ambm = xa => bm = xa1 – m . (ab)m = by => am bm = by => am = b1 – m y. Consider bm + 1 = bm b = xa1 – m b = xba1 – m = xaa1 – m = xa1 – m a = bm a . am + 1 = am a = b1 – m ya = b1 – m ay = b1 – m by = b1 – m yb = am b. Now for m > 1, b2m – 2 ab = bm – 2 bm + 1 a = bm – 2 bm aa = b2m – 2 a2 = (bm – 1 a)2 . b2m – 2 ab =
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bm – 1 bm a = bm – 1 bm + 1 = (bm)2. So (bm – 1 a)2 = (bm)2 => bm – 1 a = bm. For m = 2, b2 = ba. Similarly we prove that a2 = ab => a2 = ab, b2 = ba => a = b and a2 = ba, b2 = ab => a = b. (S, .) is weakly cancellative. Theorem 4.5: If (S, .) is weakly commutative and permutable then (S, .) is cancellative. Proof: Given that (S, .) is to be weakly commutative if for any a, b in S we have (ab)k = xa = bx for some x in S and a positive integer k. Given that (S, .) is permutable. To prove that (S, .) is cancellative. Let xa = xb. Consider (ab)k = xa => ak bk = xa => bk = xa1 – k. (ab)k = bx => ak bk = bx => ak = xb1 – k. Consider bk + 1 = bkb = xa1 – k b = a1 – k xb = a1 – k xa = xa 1 – k a = bk a. ak + 1 = ak a = xb1 – k a = b1 – k xa = b1 – k xb = xb1 – k b = ak b. Now for m > 1, b2k – 2 b2 = bk – 2 bk b2 = bk – 2 xa1 – k b2 = bk – 2 a1 – k xbb = bk – 2 a1 – k xab = bk – 2 a1 – k xba = bk – 2 a1 – k xaa = bk – 2 xa1 – k a2 = bk – 2 a2 bk = a2 b2k – 2 = (abk – 1)2. b2k – 2 b2 = bk – 2 bk b2 = bk – 2 bk + 2 = (bk )2. So (abk – 1)2 = (bk )2 => abk – 1 = bk. For k = 2, b2 = ab. Similarly we prove that a2 = ab => a2 = ab = b2 => b2 = a2 => a = b => (S, .) is cancellative. Theorem 4.6: If (S, .) is weakly commutative and permutable and separative then (S, .) is cancellative. Proof: Given that (S, .) is to be weakly commutative, permutable and separative. To prove that (S, .) is cancellative. Let xa = xb. Consider (ab)k = xa => ak bk = xa => bk = xa1 – k . (ab)k = bx => ak bk = bx => ak = xb1 – k. Consider bk + 1 = bk b = xa1 – k b = a1 – k xb = a1 – k xa = xa1 – k a = bk a . ak + 1 = ak a = xb1 – k a = b1 – k xa = b1 – k xb = xb1 – k b = a k b. Now for m > 1, b2k – 2 ab = bk – 2 bk + 1 a = bk – 2 bk aa = b2k – 2 a2 = (bk – 1 a)2 => b2k – 2 ab = bk – 1 bk a = bk – 1 bk + 1 = (bk)2 . So (bk – 1 a)2 = (bk )2 => bk – 1 a = bk. For k = 2, b2 = ba. Similarly we prove that a2 = ab. a2 = ab, b2 = ba => a = b. Similarly we prove that a2 = ba, b2 = ab => a = b. (S, .) is cancellative. REFERENCES [1] Bedrich Pondelicek, (1973), “On Weakly Commutative Semigroups” . [2 ] Chacron M., and Thierrin G., (1972), “σ-Reflexive Semigroup and Rings”, Can. Math. Bull., 15, 185-88. [3] N. P. Mukherjee, (1972), “Quasi Commutative Semigroups I”, Czech Slovak Math. Journal., 22, 449-453. [4] E. Hewitt, and H. S. Zuckerman, “The L1-Algebra of Commutative Semigroup”
