Unbounded Star Convergence in Lattices

Let L be a vector lattice, "(" x_α ") " be a L-valued net, and x∈L . If |x_α-x|∧u→┴o 0 for every u ∈〖 L〗_+ then it is said that the net "(" x_α ")" unbounded order converges to x and is denoted by □(x_α □(→┴uo x)) . This definition of unbounded order convergence has been extensively studied on many structures, including vector lattices, local solid vector lattices, normed lattices and lattice normed spaces. It is not possible to apply this type of convergence to general lattices due to the lack of algebraic structure. Therefore, we will use a type of convergence that is considered to be the motivation for this type of convergence, first defined as independent convergence in semi-ordered linear spaces and later called unbounded order convergence. Namely, L is a lattice, x_α is an L -valued net, and x ϵ L . If (x_α∧b )∨a order converges to (x∧b )∨a for every a,b∈L with a≤b, then it is said that "(" x_α ")" individual converges to x or unbounded order converges to x . This definition can be easily applied to general lattices. In this article, this definition will be understood as unbounded order convergence. Also, even if these two convergences are called by the same name, there is no equivalence between them for general lattices, an example of this is mentioned in this article. Let L be a partially ordered set, "(" x_α ")" be an L -valued net and x∈L (x_α) is said to be star convergent to x if every subnet of the net (x_α ) has a subnet that is order convergent to x and denoted by x_α □(→┴s x). In this paper, a new type of convergence on lattices is defined by combining unbounded order convergence (individual convergence) and star convergence. Let L be a lattice, (x_α ) a net and x∈L (x_α) is said to be unbounded star convergent to x if for every subnet (x_β) of (x_α), there exists a subnet (x_ζ) of (x_β) such that (x_ζ∧b)∨ □(a→┴o ) (x∧b)∨a for every a,b∈L with a≤b and it is denoted by x_α □(→┴us x). The differences between the new type of convergence, called unbounded star convergence, and order convergence, star convergence are demonstrated with counterexamples. The meaningfulness of the unbounded star convergence type is analyzed with these counterexamples and the implications presented. In addition, basic questions about unbounded star convergence of a given net on lattices such as convergence of a fixed net, uniqueness of the limit, convergence of the subnet of a convergent net are answered.