Lattice structures of fixed points of the lower approximations of two types of covering-based rough sets

Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. In this paper, we propose two family of sets and study the conditions that these two sets become some lattice structures. These two sets are consisted by the fixed point of the lower approximations of the first type and the sixth type of covering-based rough sets, respectively. These two sets are called the fixed point set of neighborhoods and the fixed point set of covering, respectively. First, for any covering, the fixed point set of neighborhoods is a complete and distributive lattice, at the same time, it is also a double p-algebra. Especially, when the neighborhood forms a partition of the universe, the fixed point set of neighborhoods is both a boolean lattice and a double Stone algebra. Second, for any covering, the fixed point set of covering is a complete lattice.When the covering is unary, the fixed point set of covering becomes a distributive lattice and a double p-algebra. a distributive lattice and a double p-algebra when the covering is unary. Especially, when the reduction of the covering forms a partition of the universe, the fixed point set of covering is both a boolean lattice and a double Stone algebra.

💡 Research Summary

The paper investigates the algebraic and order‑theoretic properties of two families of sets that arise as fixed points of lower approximation operators in covering‑based rough set theory. Coverings generalize partitions by allowing overlapping subsets of a universe U; each element x has a minimal description Md(x) and a neighborhood N(x)=⋂{K∈C | x∈K}. Two lower approximation operators are considered:

1. First‑type lower approximation (F_L): for any X⊆U, (F_L(X)=\bigcup{K∈C | K⊆X}).
2. Sixth‑type lower approximation (X_L): for any X⊆U, (X_L(X)={x∈U | N(x)⊆X}).

For each operator the authors define a fixed‑point set:

• (P_C={X⊆U | X_L(X)=X}) (fixed points of the sixth‑type, called the fixed‑point set of neighborhoods).
• (Q_C={X⊆U | F_L(X)=X}) (fixed points of the first‑type, called the fixed‑point set of covering).

The main contributions are structural theorems that describe when these families become well‑known lattice‑theoretic structures.

Fixed‑point set of neighborhoods (P_C)

• Equivalence with reduction: (P_C) coincides with the fixed‑point set obtained from the reduction of the covering (the irreducible sub‑covering). Hence redundant covering blocks do not affect the structure.
• Complete distributive lattice: For any covering C, the collection (P_C) ordered by inclusion is a lattice where join and meet are simply union and intersection, respectively. Both ∅ and U serve as the bottom and top elements, making the lattice bounded and complete.
• Double p‑algebra: Each X∈P_C possesses a pseudocomplement (X^{*}=(U\setminus X)_L) (the lower approximation of its complement) and a dual pseudocomplement (X^{+}=⋃{N(x) | x∈U\setminus X}). Consequently, (P_C) is a pseudocomplemented and dual‑pseudocomplemented lattice, i.e., a double p‑algebra.
• Boolean and double Stone algebra: When the neighborhoods ({N(x)\mid x∈U}) form a partition of U, the pseudocomplements become true complements, and the Stone identities (x^{*}∨x^{**}=1) and (x^{+}∧x^{++}=0) hold. Thus (P_C) is simultaneously a Boolean lattice, a Stone algebra, a dual Stone algebra, and therefore a double Stone algebra.

Fixed‑point set of covering (Q_C)

• Equivalence with reduction: As with P_C, the fixed‑point set does not change after removing reducible covering blocks.
• Complete lattice: For any covering, Q_C is a lattice under inclusion. The join of X and Y is X∪Y, while the meet is (F_L(X∩Y)). The lattice is bounded by ∅ and U and is complete because arbitrary meets and joins exist (via repeated application of the operators).
• Unary coverings: A covering is unary if each element of U has exactly one minimal description. Under this condition, the meet simplifies to ordinary intersection, making Q_C a distributive lattice. Moreover, the same pseudocomplement and dual pseudocomplement constructions as for P_C turn Q_C into a double p‑algebra.
• Boolean and double Stone algebra: If the reduction of C is a partition of U, then Q_C inherits the Boolean and double Stone properties, exactly as in the neighborhood case.

Methodology

The authors prove the lattice properties by first characterizing membership in the fixed‑point sets: an X belongs to P_C iff X equals the union of neighborhoods of its elements, and X belongs to Q_C iff X equals the union of all covering blocks contained in X. Using these characterizations they show closure under union and (appropriately defined) intersection. Join‑irreducible elements of the lattices are identified with the original covering blocks (or neighborhoods), which links the algebraic structure directly to the underlying data representation.

Significance

The results provide a rigorous algebraic framework for reasoning about covering‑based rough sets. By establishing when the fixed‑point families form Boolean or double Stone algebras, the paper opens the door to applying well‑developed logical and algebraic tools (e.g., Boolean algebraic simplifications, Stone duality) to problems involving overlapping or incomplete information. This has potential impact on data mining, knowledge discovery, and reasoning under uncertainty, where coverings naturally model feature subsets, overlapping clusters, or incomplete attribute information. Moreover, the identification of conditions (unary coverings, partition reductions) that guarantee stronger algebraic properties offers practical guidance for designing coverings that facilitate efficient computation and logical analysis.

Future work may explore algorithmic exploitation of these lattice structures, extensions to multi‑granular or hierarchical coverings, and connections with other generalized rough set models such as fuzzy or probabilistic coverings.