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The constructions for the unified framework of 0-overlap and 1-grouping functions on complete lattices
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DOI：10.1016/j.fss.2026.110028.png)
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Abstract
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In this paper, we study generalized overlap-grouping functions on complete lattices so as to lift the unified framework of 0-overlap functions and 1-grouping functions from the setting of [0,1] to that of complete lattices. First, we provide a method for constructing generalized overlap-grouping functions on complete lattices. Second, we prove that the ordinal sum of generalized overlap-grouping functions defined on subintervals of a meet-continuous lattice constitutes a generalized overlap-grouping function, with the boundary element pair formed by the least and greatest elements, on the sublattice consisting of all elements that are comparable to the endpoints of all subintervals. We then apply this construction method to the whole meet-continuous lattice under the additional condition that the greatest element is compact. Finally, we consider the ordinal sum construction for generalized overlap-grouping functions in the general case.
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