Finite AG‐groupoid with left identity and left zero

Abstract

Abstract. A groupoid G whose elements satisfy the left invertive law: (ab)c = (cb)a is known as Abel-Grassman's groupoid (AG-groupoid). It is a nonassociative algebraic structure midway between a groupoid and a commutative semigroup. In this note, we show that if G is a finite AG-groupoid with a left zero then, under certain conditions, G without the left zero element is a commutative group.2000 Mathematics Subject Classification. 20N99. 1.Preliminaries. An Abel-Grassman's groupoid [6], abbreviated as AG-groupoid, is a groupoid G whose elements satisfy the left invertive law: (ab)c = (cb)a. It is also called a left almost semigroup [2,3,4,5]. In [1], the same structure is called left invertive groupoid. In this note we call it AG-groupoid.It is a nonassociative algebraic structure midway between a groupoid and a commutative semigroup. The structure is medial [5], that is, (ab)(cd) = (ac)(bd) for all a, b, c, d ∈ G. It has been shown in [5] that if an AG-groupoid contains a left identity then it is unique. It has been proved also that an AG-groupoid with right identity is a commutative monoid, that is, a semigroup with identity element. An element a 0 ofLet a, b, c, and d belong to an AG-groupoid with left identity and ab = cd. Then it has been shown in [5] In this note, we show that if G is a finite AG-groupoid with left identity and a left zero a 0 , under certain conditions G\{a 0 } is a commutative group without a left zero. Results.We need the following theorem from [4] for our main result. We now state and prove our main result. Proof. Suppose that G = {a 0 ,a 1 ,...,a m }, where m is a positive integer, is an AGgroupoid with left identity under the binary operation (•). Let e be the identity element of G. It is certainly different from a 0 because of (ii) and because a 0 is the left zero under (•). The left invertive law together with (iv) implies that (aNow consider the subset G 0 of G which is obtained from it by deleting a 0 , so that Because of (iii), equation (2.3) and the facts that a −1 r is the inverse of a r under ( * ). Proof. The proof is analogous to the proof of Theorem 2.2.Acknowledgement. The authors are grateful to the referee for his invaluable suggestions.

Completely inverse AG ∗∗-groupoids

“…Moreover, if an AG-groupoid A with a left zero z is finite, then (under certain conditions) A \ {z} is a commutative group (cf. [6]).…”

Corrections and Extensions in Left and Right Almost Semigroups

“…Later in 1984 Jazek and Kepka [7] developed results on almost semigroup. In [7,16] LA-Semigroup is known as right modular groupoid, RA-Semigroup is known as left modular groupoid almost semigroup is known as bimodular groupoid. Seguier in 1904 [19] used the term semigroup for the first time in literature which is an algebraic structure S that holds associative law i.e.…”

“…Proved Results-1 in [8,14,16]: If groupoid S holds: (a) left invertive law and associative law then S is commutative semigroup as well as RA-Semigroup. (b) right invertive law and associative law then S is commutative semigroup as well as LA-Semigroup.…”

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

“…An AG-groupoid is a non-associative algebraic structure and many features of the AG-groupoid can be studied in [13]. In [14][15][16][17][18][19][20][21], some properties and connections of AG-groupoid, with some classes of algebraic structures, have been investigated. An AG-groupoid is called an AG-group if the left identity and inverse exists, while further research on the AG-group can be found in [22].…”