Interactive mathematical research suite for Admissible Partition Ideals, WQO Stabilizer Families, and \(le\)-/\(poe\)-semigroup structures. v.2026.10.23.20:49
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):
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:
👉 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.
Click a block or set in the graph to inspect its word elements, origin formula, and \(\ll\) reduction relationships.
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 \(S \in \mathbb{E}\):
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 \(\langle a \rangle_{\mathcal{P}}\):
Step-by-step instructions for exploring ordered semigroups and partition ideals.
01, 10) to compute the admissible \(\mathcal{A}\)-ideal and the Theorem 5.4.12 principal generator \(\langle a \rangle_{\mathcal{P}}\).Interactive structure explorer and algebraic ideal analyzer for \(le\)-semigroups and \(poe\)-semigroups.
An element \(a \in S\) is a \(\mathcal{P}\)-ideal element if \(a^2 \le a\) and for every block \(B \in \mathcal{P}\), \(\bigwedge_{\alpha \in B} \bar{\alpha}(a) \le a\).
01, 10 ; 010). Words in the same block are comma-separated.Find semigroups having elements \(a\) that are \(\mathcal{P}_1\)-ideal elements but NOT \(\mathcal{P}_2\)-ideal elements (\(\mathcal{I}_{\mathcal{P}_1} \setminus \mathcal{I}_{\mathcal{P}_2} \neq \emptyset\)).
An \(le\)-semigroup \(S\) is \(\mathcal{P}\)-simple if the top element \(e\) is its only \(\mathcal{P}\)-ideal element (\(\mathcal{I}_{\mathcal{P}}(S) = \{e\}\)).
01, 10 ; 010). Words in the same block are comma-separated.An element \(a\) in an \(le\)-semigroup is \(\mathcal{R}\)-regular with respect to a set of full words \(R\) if \(a \le \bar{\alpha}(a)\) for every word \(\alpha \in R\).
In partially ordered semigroups with top element \(e\), an element \(a\) is an \(\alpha\)-ideal element if \(a^2 \le a\) and \(\bar{\alpha}(a) \le a\). The semigroup is \(X\)-coincident if \(\mathcal{I}_\alpha(S) = \mathcal{I}_\beta(S)\) for all \(\alpha, \beta \in X\).
Enter words for set \(X\) (comma-separated full words):
Search entire POE dataset for structures demonstrating distinguishing behavior between two coincidence assignments.