Research Journal of Recent Sciences ______ ______________________________ ______ ____ ___ ISSN 2277 - 2502 Vol. 2 ( 3 ), 64 - 66 , March (201 3 ) Res. J. Recent Sci. International Science Congress Association 64 Short Communication Some New Results on T 1 , T 2 and T 4 - AG - groupoids Ahmad I. 1 , Rashad M. 1 and Shah M. 2 1 Department of Mathematics , University of Malakand, Khyber Pakhtunkhwa, PAKISTAN 2 Department of Mathematics, Government Post Graduate College, Mardan, PAKISTAN Available online at: www.isca.in Received 24 th September 2012 , revised 27 th October 2012 , accepted 7 th November 2012 Abstract In this article we investigate some basic properties of newly discovered classes of AG - groupoid. We consider three classes that include 1 , 2 and 4 - AG - groupoids. We prove that every 4 - AG - groupoid is Bol* - AG - groupoid. We further investigate that 1 and 4 - AG - groupoids are paramedial and hence are left nuclear square AG - groupoids. We also prove that 1 and 4 are transitively commutative AG - groupoids and 1 - AG - 3 - band is a semigroup. Keywords: AG - groupoid, LA - semigroup, AG - group, types of AG - gro upoid, nuclear square, 1 , 2 and 4 - AG - groupoids . Introduction A groupoid is called an AG - groupoid if it satisfies the left invertive law 1 : = . This structure is also known as left almost semi - group 2 (LA - semigroup), left invertive groupoid 3 and right modular groupoid 3 . An AG - groupoid is called AG - 3 - band if = = ∀ ∈ . In this paper we are going to investigate some interesting properties of newly discovered class es of namely: 1 , 2 and 4 AG - groupoids 4 . An AG - groupoid always satisfies the medial law 1 : = , while an AG - groupoid with left identity always satisfies paramedial law: = . An AG - groupoid is called transitive ly - commutative 4 if : ∀ a , b , c ∈ S , ab = ba , bc = cb implies ac = ca . Recently some new classes of AG - groupoid have been discovered 5, 6 that are; Bol*, 1 , 2 , 3 and 4 - AG - groupoid and some others. Here we consider 1 , 2 and 4 - AG - groupoids to further investigate them. An AG - groupoid is called 1 - AG - groupoid if ∀ , , , ∈ , = = and is called 2 - AG - groupoid if ∀ , , , ∈ , = = . An AG - groupoid is called forward 4 - AG - groupoid denoted by 4 if ∀ , , , ∈ , = = , and is called backward 4 - AG - groupoid, denoted by 4 , if ∀ , , , ∈ , = = . An AG - groupoid is called 4 - AG - groupoid if it is both forward and backward 4 - AG - groupoid. An AG - groupoid is called left nuclear square 4 if ∀ , , ∈ , 2 = 2 . Right nuclear and nuclear square can be defined analogously. A groupoid is called Bol* - groupoid if it satisfies the identity . = . . A groupoid S is called left cancellative 4 if ∀ , , ∈ , = = . Right cancellative and cancellative AG - groupoid can be defined similar ly. It should be noted that various algebraic structures can be constructed from each other by defying a suitable relation between them. A similar article 7 by the authors can be seen that how some new classes of AG - groupoids can be constructed from other known classes. A generalization of cancellative AG - groupoid has been done as quasi - cancellativity 8 . AG - groupoids; generalize commutative semigroups and have applications in flock theory 9 and some in geometry 4 . Properties of , and - AG - gro upoids It is proved that every 1 - AG - groupoid is Bol* - AG - groupoid 5 , and that every Bol* - AG - groupoid is paramedial AG - groupoid 6 . Here we proceed to prove that every 1 - AG - groupoid is paramedial but the converse is not true. Theorem 1 : Every 1 - AG - groupoid is paramedial - AG - groupoid Proof. Let be a 1 - AG - groupoid, and let , , , ∈ . Then by definition of 1 - AG - groupoid = ⇒ = . Now since, . = . ⇒ . = . 1 − − = . ⇒ . . ( 1 − − ) = . ⇒ . = . . Hence is paramedial - AG - groupoid. Here is an example of paramedial AG - groupoid that is not - AG - groupoid. Example 1. Paramedial AG - groupoid of order 3 which is not 1 - AG - groupoid. Research Journal of Recent Sciences ______ _ _ _______________________________ ______________ _ ________ ISSN 2277 - 2502 Vol. 2 ( 3 ), 64 - 66 , March (201 3 ) Res. J. Recent Sci. International Science Congress Association 65 * 1 2 3 1 1 1 1 2 1 1 1 3 2 2 2 Since each paramedial is left nuclear sqauare 4 . The following corollary is now an obvious fact. Corollary 1 : Every 1 - AG - groupoid is left nuclear square AG - groupoid. Theorem 2. Bol * - AG - groupoid with left identity is 1 - AG - groupoid. Proof. Let be a Bol * - AG - groupoid with left identity , and , , , ∈ . Let = . Then, = . = . ∗ − − = = = . ( ) = . ( ) ⇒ = . Hence is 1 - AG - groupoid. Corollary 2 . Bol * - AG - groupoid with left identity is 3 - AG - groupoid. Theorem 3. Every 1 - AG - 3 - band is semigroup. Proof. Let be 1 - AG - groupoid that is also AG - 3 - band. Then ∀ , , , ∈ ; = ⇒ = Now since, = ( ) ⇒ = 1 − − = ( − 3 − ) = . ( ) ⇒ = ( ) . ⇒ = . ( 1 − − ) = . = ( − 3 − ) ⇒ . Hence is a semigroup 10 . Theorem 4 : Every 2 - AG - groupoid is transitively commutative AG - groupoid. Proof : Let be 2 - AG - groupoid. Then ∀ , , , ∈ , we have = ⇒ = Let = , = . Consider = (1) ⇒ = ( 2 − − ) ⇒ = ( = ) ⇒ = ( 2 − − ) ⇒ = ( = ) ⇒ = ( 2 − − ) (2) ⇒ = ( 1 2 ) Hence is transitively commutative AG - groupoid. Theorem 5 : Let be an AG - groupoid with left identity such that 2 = ∀ ∈ . Then is 2 - AG - groupoid. Proof. Let be an AG - monoid with left identity such that 2 = . Let , , , ∈ and = . (3) Then, = = ( by left invertive law ) = . ( by assumption ) = . ( by medial law ) = . by Equation 3 = . by medial law = . by assumption ⇒ = . ( ) Hence is 2 - AG - groupoid. We will use the following lemmas to prove some further properties of 2 - AG - groupoid. Lemma 1 : Every 1 - AG - groupoid is Bol* - AG - groupoid 4 . Lemma 2 : Every 2 - AG - groupoid is 1 - AG - groupoid 4 . Since, by Lemma 1 and 2, we know that every 1 - AG - groupoid is Bol* - AG - groupoid and every 2 - AG - groupoid is 1 - AG - groupoid. We immediately have the following result. Corollary 3 : Every 2 - AG - groupoid is Bol* - AG - groupoid. Proof : Let be a 2 - AG - groupoid. Then ∀ , , , ∈ we have = ⇒ = . Now consider, . . (4) ⇒ . = . 2 = . ( ) ⇒ . = . ( 2 − − ) . = . ( 2 ) ⇒ . = . ( 2 − − ) ⇒ . = . ( 2 ) ⇒ . = . ( ) (5) ⇒ . = . ( 4 5 ) ⇒ . = . ( ) Hence is Bol* - AG - groupoid. Since every 2 - AG - groupoid is 1 - AG - groupoid 4 and every 1 - AG - groupoid is paramedial AG - groupoid by Theorem 1 and is left nuclear square by Corollary 1, thus we have the following: Research Journal of Recent Sciences ______ _ _ _______________________________ ______________ _ ________ ISSN 2277 - 2502 Vol. 2 ( 3 ), 64 - 66 , March (201 3 ) Res. J. Recent Sci. International Science Congress Association 66 Corollary 4 : Every 2 - AG - groupoid is paramedial AG - groupoid. Corollary 5 : Every 2 - AG - groupoid is left nuclear square AG - groupoid. The following result gives an interesting relation between 4 - AG - groupoids and Bol* - AG - groupoids. Theorem 6 : Every 4 - AG - groupoid is Bol* - AG - groupoid. Proof : Let be 4 - AG - groupoid, and let , , , ∈ . Then by definition 4 - AG - groupoid = ⇒ = ( 4 − − ) = ⇒ = ( 4 − − ) Now let, . = . ( ) ⇒ . = . ( 4 − − ⇒ . = . ( 4 − − ) = . ( ) ⇒ . = ( . ) ( . ) ( 4 − − ) ⇒ . = ( . ) ( 4 − − ) = . ⇒ . = . . Hence is Bol* - AG - groupoid. Since each Bol * - AG - groupoid is parmedial 4 and thus is left nuclear square 4 , whence using Theorem 6 we immediately have the following: Corollary 6 : Every 4 - AG - groupoid is paramedial AG - groupoid. Corollary 7 : Every 4 - AG - groupoid is left nuclear square AG - groupoid. Next we prove that the class of transitively commutative AG - groupoids contains the class of all 4 - AG - groupoid s. Theorem 7 : Every 4 - AG - groupoid is transitively commutative AG - groupoid. Proof. Let be 4 - AG - groupoid. Then ∀ , , , ∈ , we have = ⇒ = ( 4 − − ) , and = ⇒ = ( 4 − − ) Now let, = and = . Consider = and = ⇒ = . Applying definition of 4 - AG - groupoid, we have, = . Hence is transitively commutative AG - groupoid. It is known that every 1 - AG - groupoid is ∗ ∗ - groupoid 11 . Here we prove that every c ancellative ∗ ∗ - groupoid is 1 - AG - groupoid. Theorem 8 : Every cancellative ∗ ∗ - groupoid is 1 - AG - groupoid. Proof : Let be a cancellative ∗ ∗ - groupoid and let be a cancellative element of . Then ∀ , , , ∈ , let = . Now since, 2 = 2 ∗ ∗ − = . = . ∗ ∗ − = . = . = . ∗ ∗ − = . = . ∗ ∗ − = . ( ) 2 . = 2 . ⇒ = ( ) Hence is 1 - AG - groupoid. Conclusion Many new classes of AG - groupoids have been discovered recently. Enumeration has also been done of these new classes up to order 6. All this has attracted researchers of the field to investigate these newly discovered classes in detail. This current article investigates the ideas of 1 , 2 and 4 - AG - groupoids. We investigate that every 4 - AG - groupoid is Bol* - AG - groupoid. We further investigate that 1 and 4 - AG - groupoids are paramedial and hence are left nuclear square. We also prove that 1 and 4 are transitively commutative AG - groupoi d and 1 - AG - 3 - band is a semigroup. References 1. Kazim M.A. and Naseerudin M., On almost semigroups Portugalian Mathematica , 36 , 1 , ( 1977 ) 2. Holgate P., Groupoids satisfying a simple invertive law, Math. Stud., 61 , 101 - 106 (1992) 3. Cho J.R., Pusan, Jezek J. and Kepka T., Praha, Paramedial Groupoids, Czechoslovak Mathematical Journal , 49(124), (1996) 4. 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Math. Forum, 8, ( 5 ) , 237 - 243 (2013) Modified 23 rd A pril 2013