Algebraic structures of picture fuzzy level operators with metric properties

This paper investigates the algebraic structure of picture fuzzy level operators. The concept of level subsets for picture fuzzy sets was introduced by Kankaew et al. ^[1] in the context of UP (BCC)-algebras, where they defined the upper and lower level subsets $ \mathcal{U}_\mu(t) $, $ \mathcal{U}_\eta(t) $, and $ \mathcal{L}_\nu(t) $. Building upon this foundational work, we define picture fuzzy level operators $ L_{\alpha, \beta, \gamma} $ and investigate their algebraic properties. Let $ \mathcal{L} = \{ L_{\alpha, \beta, \gamma} : \alpha, \beta, \gamma \in [0, 1], \; 0 \le \alpha+\beta+\gamma \le 1 \} $ denote the admissible set of all picture fuzzy level operators, and let $ \overline{\mathcal{L}} = \{ L_{\alpha, \beta, \gamma} : \alpha, \beta, \gamma \in [0, 1] \} $ denote the extended operator space obtained by dropping the constraint $ \alpha+\beta+\gamma \le 1 $. We define a partial order $ \preceq $ on $ \overline{\mathcal{L}} $ by componentwise order with $ \gamma $ reversed. We prove that $ (\overline{\mathcal{L}}, \preceq) $ is a poset, with $ \mathcal{L} $ as a sub-poset. We introduce the meet $ \sqcap $ and join $ \sqcup $ operations and show that $ (\overline{\mathcal{L}}, \sqcap, \sqcup) $ forms a distributive lattice, and moreover a complete lattice. The admissible set $ \mathcal{L} $ is characterized as closed under meet but not under join. We define composition $ \circ $ of level operators and prove that $ (\overline{\mathcal{L}}, \circ) $ forms a commutative idempotent monoid with identity $ L_{0, 0, 1} $, with the interesting property that composition coincides with the join operation. We investigate inverse operators, proving that only the identity operator has a true inverse, and introduce dual operators satisfying natural duality properties. We establish metric structures, including Chebyshev, Manhattan, and Euclidean distances on $ \overline{\mathcal{L}} $. Finally, we provide a geometric interpretation via the operator simplex—a tetrahedral representation in $ \mathbb{R}^3 $ that visualizes the admissible set of level operators. This work provides a systematic algebraic and geometric framework for understanding picture fuzzy level operators.