By I. M. Yaglom
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Ii. K ∩ I = (0) and K1 is an S-ideal of G such that K ⊂ K1 then K1 ≠ K imply K1 ∩ I ≠ (0). 55: A subset S of the S-N-subsemigroup G is said to be Smarandache small (S-small) in G if i. ii. iii. S + K = G where K is an S-ideal of G imply K = G. G is said to be Smarandache hollow (S-hollow) if every proper S-ideal of G is S-small in G and G is said to have Smarandache finite spanning dimension (S-finite spanning dimension) if for any decreasing sequence of Smarandache NS-subsemigroups (S-NS-subsemigroups); X ⊃ X1 ⊃…of G such that Xi is an S-ideal of Xi-1 there exists an integer such that Xj is S-small in G for all j ≥ k.
I. ii. iii. iv. v. vi. vii. viii. ix. x. To every pair of vectors α, β in V there is associatied a vector in V called the sum which we denote by α + β. Addition is associative (α + β) + γ = α + (β + γ) for all α , β , γ ∈ V. There exists a vector which we denote by zero such that 0 + α = α + 0 = α for all α ∈ V. e α +β = β + α, α , β ∈ V. For 0 ∈ S and α ∈ V we have 0 yα = 0. To every scalar s ∈ S and every vector v ∈ V there is associated a unique vector called the product s y v, which we denote by sv.
0 y a = a y 0 = 0 for all a ∈ D. a y (b + c) = a y b + a y c for all a, b, c ∈ D. For every pair a, b ∈ D there exists da, b ∈ D∗ such that for every x ∈ D; a + (b + x) = (a + b) + da, b x. Now [118 and 126] has defined loop near domains analogous to group rings. DEFINITION : Let L be a finite loop under ‘+’ and D be a near domain, the loop near domain DL contains elements generated by di mi where di ∈ D and mi ∈ L where we admit only finite formal sums satisfying the following: i. ii. iii.
Álgebra Extraordinaria by I. M. Yaglom