<![CDATA[Order convergence is a crucial concept in the theory of Riesz spaces. A generalization of order convergence, known as unbounded order convergence (or uo convergence), has been introduced. In this paper, we present a further generalization of uo convergence by using an arbitrary nonempty subset of the positive cone of the space. We then investigate the properties of this generalization, including uniqueness of limits, algebraic properties, and the relationships among these types of convergence. In particular, we show that the generalizations of uo convergence generated by two subsets are equivalent if and only if the bands generated by those subsets are equal.]]>
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