Verification and hunting engine for Admissible Partition Ideal Assignments, WQO Stabilizer Family \(\mathbb{E} = \min_{\ll} \mathcal{R}(\mathcal{P})\), and Principal Generator Formula \(\langle a \rangle_{\mathcal{P}}\) in \(le\)-semigroups.
1. Select Preset or Hunt Partition Ideal Assignment \(\mathcal{P}\)
According to Definition 5.4.7, \(\mathcal{P}\) is an admissible partition ideal assignment if and only if every distinct index pair \(i \neq j\) satisfies both the Concatenation Condition (Condition 1) and the Set Replacement Condition (Condition 2):
Step 3
WQO Stabilizer Family \(\mathbb{E} = \min_{\ll} \mathcal{R}(\mathcal{P})\)
By Higman's Lemma, the set of full words under the reduction preorder \(\le\) forms a well-quasi-ordering (WQO). Iterated set replacements and concatenations filtered by the set reduction preorder \(\ll\) stabilize in finite steps to the minimal stabilizer family:
🌌 Interactive WQO Hasse Diagram & Lineage Network
👉 Instructions: Click any node in the graph to inspect word elements, origins, and \(\ll\) relationships in the side panel. You can drag, zoom, pan, or click ⛶ Fullscreen Canvas.
👆
No Node Selected
Click a block or set in the graph to inspect its word elements, origin formula, and \(\ll\) reduction relationships.
Step 4
Word Valuations & Lattice Meets
Evaluating word valuations \(\overline{\omega}(a)\) (mapping \(0 \mapsto e\), \(1 \mapsto a\)) and taking the lattice meet \(\bigwedge_{\omega \in S} \overline{\omega}(a)\) over each minimal set \(S \in \mathbb{E}\):
★
Theorem 5.4.12 Principal Generator Formula
Because \(\mathcal{P}\) satisfies all admissibility conditions, the supremum join over the stabilizer family valuations yields the exact algebraic generator formula for the principal partition ideal element \(\langle a \rangle_{\mathcal{P}}\):