Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 2(4), 91-93, April (2013) Res. J. Recent Sci. International Science Congress Association 91 Some Properties of AG*-groupoid Ahmad I. 1 , Rashad M. 1 and Shah M. 2 1 Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, PAKISTAN 2 Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, PAKISTAN Available online at: www.isca.in Received 1 st November 2012, revised 2 nd December 2012, accepted 5 th January 2013 Abstract We prove some new results for AG * -groupoid such as: (1) an AG * -groupoid is a Bol * -AG-groupoid, (2) an AG * -groupoid is nuclear square AG-groupoid, (3) a cancellative AG * -groupoid is transitively commutative AG-groupoid, (4) a -AG-3-band is AG * -groupoid, (5) an AG-groupoid with a left cancellative element is a -AG-groupoid, and (6) an AG * -groupoid is left alternative AG-groupoid. Keywords: AG-groupoid, AG * -groupoid, AG-group, types of AG-groupoid, paramedial AG-groupoid, Nuclear square AG- groupoid. Introduction A groupoid (, . ) or shortly is called an Abel Grassmann groupoid or shortly AG-groupoid (also called a left almost semigroup 1 , shortly LA-semigroup) if it satisfies the left invertive law: () = (). Other names such as left invertive groupoid 2 , and right modular groupoid 3 can also be found for this structure in the literature. Clearly every commutative semigroup satisfies the left invertive law and thus becomes an AG-groupoid and so an AG-groupoid generalizes commutative semigroup. An AG-groupoid satisfying the identity () = () is AG*-groupoid. Recently some new classes 4-7 of AG-groupoid have been defined. In this paper we investigate some interesting relations of AG * -groupoid with some of those new classes. We first recall some necessary definitions before to discuss in detail the results that we have listed in the abstract. An AG-groupoid is called: −an AG-3-band if () = () = ∀ ∈ . −a transitively commutative AG-groupoid if ∀,, ∈ , = , = ⇒ = . −a -AG-groupoid if ∀ , , , ∈ , = ⇒ = . −a -AG-groupoid if ∀ , , ∈ , = ⇒ = . −a Bol*-AG-groupoid if it satisfies the identity (. ) = (. ). −a paramedial-AG-groupoid if ∀ , , , ∈ , ()() = ()(). −a left nuclear square AG-groupoid 8 if ∀ , , ∈ () = ( ). −a right nuclear square AG-groupoid 8 if ∀ , , ∈ () = ( ). −a middle nuclear square AG-groupoid 8 if ∀ , , ∈ ( ) = ( ). −a nuclear square AG-groupoid if it is left, right and middle nuclear square AG-groupoid 8 . −a left alternative AG-groupoid if ∀ , ∈ , . = . . Let be an AG-groupoid. An element of is called left cancellative if = ⇒ = ∀ , ∈ . Similarly an element of is called right cancellative if = ⇒= ∀, ∈ . An element of is called cancellative if it is both left and right cancellative. is called left cancellative (right cancellative, cancellative) if every element of is left cancellative (right cancellative, cancellative). For detailed study of cancellativity of AG-groupoids we refer the reader to reference 9 . It has been proved that a Bol * -AG-groupoid is paramedial AG-groupoid and a paramedial AG-groupoid is left nuclear square 6 . Recall that an AG-groupoid always satisfies the medial law 1 : ()() = ()(). It can be easily verified that a -AG-groupoid is a -AG-groupoid. Some Properties of AG * -groupoid We start with the following theorem that gives a relation between AG* -groupoids and Bol*-AG-groupoids. Theorem 1: Every AG*-groupoid is Bol* -AG-groupoid. Proof: Let be an AG* -groupoid, and let , , , ∈ . Then by definition of AG*-groupoid () = (). Now (. ) = . (by left invertive law) ⇒ (. ) = . (by medial law) ⇒ (. ) = (. ) (by definition of AG * -groupoid) ⇒ (. ) = (. ) (by definition of AG * -groupoid) ⇒ (. ) = (. ) (by left invertive law) ⇒ (. ) = (. ) (by definition of AG * -groupoid) ⇒ (. ) = . (by definition of AG * -groupoid) ⇒ (. ) = (. ) (by left invertive law) ⇒ (. ) = (. ) (by definition of AG * -groupoid) Hence AG* -groupoid is Bol* -AG-groupoid.