Convergence in lattice normed spaces normed by f-algebras

A net (x_α )_(α∈A) in a vector lattice E is said to be order convergent to a vector x∈E if there exists another net (y_β )_(β∈B)↓0 such that for every β, there is an index α_β such that |x_α-x|≤y_β for all indices α≥α_β and abbreviated by x_α □(→┴o ) x. A vector lattice E under an associative multiplication is said to be a Riesz algebra whenever the multiplication makes E an algebra (with the usual properties), and in addition, it satisfies the following property: x,y ∈E implies x∙y∈E_+. A Riesz algebra E is called f-algebra if E has additionally property that x∧y=0 implies (x∙z)∧y=(z∙x)∧y=0 for all z∈E_+. A net (x_α )_(α∈A) in E is said to be multiplicative order convergent to x∈E if |x_α-x|∙u□(→┴o ) 0 for all u∈E_+. Abbreviated as □(x_α →┴mo ) x. In this study, in the light of this given way, we basically defined the concept of the u_f-convergence on lattice norms and by examining its relations with the convergences given above.