Completely inverse AG ∗∗-groupoids

Abstract

A completely inverse AG

Characterizations of intra-regular Abel-Grassmann's groupoids

“…Conversely, since a ∈ Sa 2 for every a ∈ S, thus a ∈ Sa 2 = eS · a 2 = a 2 S · e ⊆ a 2 S · S = (a 2 · eS) · S (5) = (Se · a 2 ) · S ⊆ Sa 2 · S.…”

“…ab · cd = dc · ba (5) for all a, b, c, d ∈ S. Let S be an AG-groupoid. By an AG-subgroupoid of S we mean a non-empty subset A of S such that A 2 ⊆ A.…”

“…On the other hand ua = u(xa 2 · y) (4) = xa 2 · uy (5) = yu · a 2 x (4) = a 2 · (yu · x) (5) = (x · yu) · a 2…”

“…For example, congruences of some AGgroupoids have very similar properties as congruences of semigroups (cf. [5], [6] and [18]). Moreover, any locally associative AG-groupoid S with left identity is uniquely expressible as a semilattice of archimedean components.…”

“…Indeed, using(1)and(4), for every a ∈ S we getS(Sa ∩ aS) ∩ (Sa ∩ aS)S = S(Sa) ∩ S(aS) ∩ (Sa)S ∩ (aS)S = S(Sa) ∩ aS ∩ (Sa)S ∩ Sa ⊆ Sa ∩ aS,which shows that Sa ∩ aS is a quasi-ideal. Moreover, S(Sa) ∩ (Sa)S ⊆ S(Sa) = SS · Sa(5) = aS · SS = aS · S(1) = SS · a = Sa and S(Sa 2 ) ∩ (Sa 2 )S ⊆ (Sa 2 )S = Sa 2 · S 2(5) …”