Proof. Let \(S\) be a non-empty subset of \(X\). Note that \(\emptyset \neq C(S) \subseteq O\) and \((O, <_{O})\) is well-ordered. Thus, \(C(S)\) possesses a unique least element with respect to \(<_{O}\), which we denote by \(a\). If \(a \in O_{0}\), then \(a\) is the least element of \(S\) with respect to \(<_{X}\). Alternatively, assume that \(a \in O_{+}\), and define the fiber \[\begin{align} S_{a} := \{s \in S \mid C(s) = a\}. \end{align}\] Since \(\emptyset \neq S_{a} \subseteq C_{a}\) and \((C_{a}, <_{C_{a}})\) is well-ordered, \(S_{a}\) contains a least element with respect to \(<_{C_{a}}\), say \(b\). By definition, \(b\) constitutes the least element of \(S\) with respect to \(<_{X}\). Therefore, \((X, <_{X})\) is well-ordered.

Next, let \(T\) be a non-empty subset of \(\langle O \rangle\), and define the set of rightmost letters: \[\begin{align} T_{r} := \{r(\alpha) \mid \alpha \in T\}. \end{align}\] Given that \(\emptyset \neq T_{r} \subseteq O\) and \((O, <_{O})\) is well-ordered, \(T_{r}\) has a least element with respect to \(<_{O}\), say \(c\). Let \[\begin{align} T_{r,c} := \{\alpha \in T \mid r(\alpha) = c\}. \end{align}\] Among the words in \(T_{r,c}\), choose those with minimal length \(m\), and set \[\begin{align} T_{r,c,m} := \{\alpha \in T \mid r(\alpha) = c, ~|\alpha| = m\}. \end{align}\] Observe that \(\emptyset \neq T_{r,c,m} \subseteq O^{m}\) and \((O^{m}, <_{\operatorname{mlex}_{O}})\) is well-ordered. Thus, \(T_{r,c,m}\) has a unique least element with respect to \(<_{\operatorname{mlex}_{O}}\), which can be expressed as \(\beta c\) for some word \(\beta\). By definition, \(\beta c\) is the least element of \(T\) with respect to \(<_{\langle O \rangle}\), and hence \((\langle O \rangle, <_{\langle O \rangle})\) is well-ordered.

Finally, let \(U\) be a non-empty subset of \(\langle X \rangle\). Since \(\emptyset \neq C(U) \subseteq \langle O \rangle\) and \((\langle O \rangle, <_{\langle O \rangle})\) is well-ordered, \(C(U)\) has a least element with respect to \(<_{\langle O \rangle}\), say \(\gamma\). Let \[\begin{align} U_{\gamma} := \{\alpha \in \langle X \rangle\mid C(\alpha) = \gamma\}. \end{align}\] Note that \(\emptyset \neq U_{\gamma} \subseteq X^{|\gamma|}\). Since \((X, <_{X})\) is well-ordered, the Cartesian power \((X^{|\gamma|}, <_{\operatorname{mlex}_{X}})\) is also well-ordered. Then \(U_{\gamma}\) contains a unique least element with respect to \(<_{\operatorname{mlex}_{X}}\), say \(\eta\). By definition, \(\eta\) is the least element of \(U\) with respect to \(<_{\langle X \rangle}\). Therefore, \((\langle X \rangle, <_{\langle X \rangle})\) is well-ordered. ◻

Proof. Suppose for contradiction that the assertion is false. Then there exists an \(I\)-irreducible letter \(y \in O\) such that \[\begin{align} y \in \mathbb{k}\langle X \rangle\cdot X^{< y} + I. \end{align}\] Consequently, there exist polynomials \(f_{i} \in \mathbb{k}\langle X \rangle\), letters \(x_{i} \in X^{< y}\), and scalars \(k_{i} \in \mathbb{k}\) such that the polynomial \[\begin{align} p = y - \sum_{i} k_{i}f_{i}x_{i} \end{align}\] belongs to \(I\). Since the rightmost letter of each term satisfies \(r(f_{i}x_{i}) = x_{i} <_{X} y\), Corollary 8 ensures that \(C(f_{i}x_{i}) <_{\langle O \rangle} y = C(y)\) for all indices \(i\). From the definition of the reduction order on \(\langle X \rangle\), it follows that \(f_{i}x_{i} <_{\langle X \rangle} y\), which yields \(\operatorname{LW}(p) = y\). Since \(p \in I\), this implies that \(y\) is \(I\)-reducible, yielding the desired contradiction. ◻

Proof. Suppose for contradiction that \(O_{I}\) is infinite. We can therefore choose an infinite sequence of distinct letters \(x_{1}, x_{2}, \ldots, x_{n}, \ldots\) in \(O_{I}\) ordered such that \[x_{1} <_{O} x_{2} <_{O} \ldots <_{O} x_{n} <_{O} \ldots.\] Using these elements, we construct the following ascending chain of left ideals in \(\mathbb{k}\langle X \rangle\): \[\begin{align} \mathbb{k}\langle X \rangle\cdot x_{1} + I \subseteq \mathbb{k}\langle X \rangle\cdot \{x_{1},x_{2}\} + I \subseteq \ldots \subseteq \mathbb{k}\langle X \rangle\cdot \{x_{1},\ldots,x_{n}\} + I \subseteq \ldots. \end{align}\] Since \(X, I \subseteq \ker \epsilon\), it follows that \(1 \notin \mathbb{k}\langle X \rangle\cdot \{x_{1},\ldots,x_{n}\} + I\), ensuring that each left ideal in the chain is proper. Furthermore, Lemma 23 implies that \[\begin{align} x_{n+1} \notin \mathbb{k}\langle X \rangle\cdot \{x_{1},\ldots,x_{n}\} + I \qquad \text{for all } n \ge 1. \end{align}\] Consequently, the chain of left ideals is strictly ascending.

Since \(\pi\) is an algebra homomorphism, applying \(\pi\) to this sequence yields a strictly ascending chain of left ideals in \(A\): \[\begin{align} A \cdot \pi(x_{1}) \subsetneq A \cdot \{\pi(x_{1}),\pi(x_{2})\} \subsetneq \ldots \subsetneq A \cdot \{\pi(x_{1}),\ldots,\pi(x_{n})\} \subsetneq \ldots. \end{align}\] This directly contradicts the hypothesis that \(A\) is left Noetherian. Therefore, \(O_{I}\) must be finite, which immediately implies that its subset \(O_{I,+}\) is finite. ◻

Proof. By Lemma 1, we have an isomorphism of graded Hopf algebras \(\operatorname{gr}_{\operatorname{s}} H \cong B \sharp H_{[0]}\), where \(B = \bigoplus_{n\ge 0} B_{n}\) is a graded connected Hopf algebra in the Yetter-Drinfel’d category \({^{H_{[0]}}_{H_{[0]}}\mathcal{YD}}\), and \((\operatorname{gr}_{\operatorname{s}} H)_{n} = B_{n}H_{[0]}\).

For each \(b \in B_{n}\) with \(n \ge 1\), there exist elements \(b_{1} \in B_{n-k}\) and \(b_{2} \in B_{k}\) (\(1 \le k \le n\)) such that \[\begin{align} \Delta_{B}(b) = b \otimes 1 + 1 \otimes b + \sum b_{1} \otimes b_{2}. \end{align}\] Consequently, there exist elements \(b_{(-1)}, b_{2(-1)} \in H_{[0]}\), \(b_{(0)} \in B_{n}\), and \(b_{2(0)} \in B_{k}\) (\(1 \le k \le n\)) such that \[\begin{align} \Delta_{\operatorname{gr}_{\operatorname{s}} H}(b) = b \otimes 1 + \sum b_{(-1)} \otimes b_{(0)} + \sum \sum b_{1}b_{2(-1)} \otimes b_{2(0)}. \end{align}\] Applying the counit axiom for \(H_{[0]}\)-comodules, we can express \(\rho_{B}(b) = (1-z_{b}) \otimes b + v\) for some elements \(z_{b} \in H_{[0]} \cap \ker \epsilon_{H}\) and \(v \in (H_{[0]} \cap \ker \epsilon_{H}) \otimes (\operatorname{gr}_{\operatorname{s}} H)_{n}\). Hence, we can write \[\begin{align} \Delta_{\operatorname{gr}_{\operatorname{s}} H}(b) = b \otimes 1 + 1 \otimes b + \sum b' \otimes b'' \end{align}\] for some \(b' \in (\operatorname{gr}_{\operatorname{s}} H)_{n-k} = B_{n-k}H_{[0]}\) and \(b'' \in (\operatorname{gr}_{\operatorname{s}} H)_{k} = B_{k}H_{[0]}\) (\(1 \le k \le n\)).

Next, define the homogeneous filtration components: \[\begin{align} H_{B,n} &:= \{ d \in H_{[n]} \mid 0 \neq d + H_{[n-1]} \in B_{n} \}, \quad n \ge 1, \qquad H_{B} := \bigcup_{n \ge 1} H_{B,n}. \end{align}\] It is straightforward to verify that \[\begin{align} H_{[n]} = \mathbb{k}H_{B,n}H_{[0]} + H_{[n-1]}, \qquad H = \mathbb{k}H_{B}H_{[0]} + H_{[0]}. \end{align}\] Thus, for each \(d \in H_{B,n} \cap \ker \epsilon_{H}\), there exist elements \(d'_{+} \in \bigcup_{k=1}^{n-1} H_{B,k} \cup \{1\}\), \(d'_{0} \in H_{[0]}\), and \(d'' \in H_{[n]} \cap \ker\epsilon_{H}\) such that \[\begin{align} \Delta_{H}(d) = d \otimes 1 + 1 \otimes d + \sum d'_{+} d'_{0} \otimes d'', \quad \text{with } d'_{+}d'_{0} \in \ker \epsilon_{H}. \end{align}\]

Let \(L\) be a basis of \(H_{[0]}\) such that \(\epsilon_{H}(b) \in \{0, 1\}\) for each \(b \in L\), and set \(L' := \{\epsilon_{H}(b)1 - b \mid b \in L\}\). One can verify that for each \(z \in L'\), there exist elements \(z' \in L'\) and \(r_{z}, z'' \in \mathbb{k}L'\) such that \[\begin{align} \Delta_{H}(z) = z \otimes (1-r_{z}) + 1 \otimes z + \sum z' \otimes z'', \quad \text{with } z' \neq z. \end{align}\]

We now choose a set \(O\) of formal variables to serve as a generating set for \(H\), equipped with an assignment map \(f_{O}: O \rightarrow H\) and a degree function \(t_{O}: O \rightarrow \mathbb{N}\) defined by \(t_{O}(y) = \min\{n \mid f_{O}(y) \in H_{[n]}\}\). Here, \(O_{0}\) denotes the set of formal variables corresponding to \(L'\), chosen such that \[\begin{align} f_{O}(O) \subseteq \ker \epsilon_{H} \quad \text{and} \quad f_{O}(O_{0}) = L'. \end{align}\] We define the expanded variable sets: \[\begin{align} X &:= O_{0} \cup \left(\bigcup_{y\in O_{+}} C_{y}\right), \qquad C_{y} := \{y_{a} \mid a \in O_{0}^{*}\} \quad \text{for } y \in O_{+}, \end{align}\] and define the assignment \(f_{X}: X \rightarrow H\) via \[\begin{align} f_{X}(x) &= f_{O}(x), \qquad \text{for } x \in O,\\ f_{X}(y_{a}) &= \sum f_{O}(a)_{(1)} f_{O}(y) S_{H}(f_{O}(a)_{(2)}), \qquad \text{for } y_{a} \in C_{y}, ~y \in O_{+}, ~ a \in O_{0}. \end{align}\] Evidently, \(f_{X}(X) \subseteq \ker \epsilon_{H}\). Let \(\pi: \mathbb{k}\langle X \rangle\rightarrow H\) be the canonical algebra projection lifted from \(f_{X}\), and let \(I := \ker \pi\). Recall that the induced map \(t_{X}: X \rightarrow \mathbb{N}\) satisfies \(t_{X}(a) = t_{O}(a)\) and \(t_{X}(y_{b}) = t_{X}(y) = t_{O}(y)\) for \(a,b \in O_{0}\) and \(y \in O_{+}\). It follows that \(X_{0} = O_{0}\), and the free algebra \(\mathbb{k}\langle X \rangle\) inherits the grading \(\mathbb{k}\langle X \rangle= \bigoplus_{n\ge 0} \mathbb{k}\langle X \rangle_{n}\) determined by \(\deg(x) = t_{X}(x)\) for \(x \in X\).

By the local finiteness of coalgebras, the generating sets \(O\) and \(X\) can be chosen such that \(\Delta_{H}\) lifts to an algebra homomorphism \(\Delta: \mathbb{k}\langle X \rangle\rightarrow \mathbb{k}\langle X \rangle\otimes \mathbb{k}\langle X \rangle\) satisfying \(\Delta_{H} \circ \pi = (\pi \otimes \pi) \circ \Delta\), subject to the following conditions:

• For every \(x \in X_{0}\), there exist letters \(x' \in X_{0}\) and polynomials \(r_{x}, x'' \in \mathbb{k}\langle X_{0} \rangle^{+}\) such that \[\begin{align} \Delta(x) = x \otimes (1-r_{x}) + 1 \otimes x + \sum x' \otimes x'', \quad \text{with } x' \neq x; \end{align}\]

• For every \(x \in X_{+}\), there exist elements \(x'_{+} \in X_{+}^{*}\), \(x'_{0} \in X_{0}^{*}\), and \(x'' \in \mathbb{k}\langle X \rangle^{+}\) such that \[\begin{align} \Delta(x) = x \otimes 1 + 1 \otimes x + \sum x'_{+}x'_{0} \otimes x'', \quad \text{with } x'_{+}x'_{0} \neq 1 \text{ and } \;t_{X}(x'_{+}) < t_{X}(x). \end{align}\]

By construction, \(\Delta\) defines a comultiplication on \(\mathbb{k}\langle X \rangle\) such that \(\Delta(I) \subseteq I \otimes \mathbb{k}\langle X \rangle+ \mathbb{k}\langle X \rangle\otimes I\). Analogously, the counit \(\epsilon_{H}\) lifts to an algebra map \(\epsilon: \mathbb{k}\langle X \rangle\rightarrow \mathbb{k}\) satisfying \(X, I \subseteq \ker \epsilon\).

Finally, we impose a total order \(<_{X}\) on \(X\). We first fix a well-ordering \(<_{O_{n}}\) on each homogeneous component \(O_{n}\) for \(n \ge 0\). For any \(x,y \in O\), we define: \[\begin{align} x <_{O} y \Longleftrightarrow \left\{\begin{array}{l} t_{O}(x) < t_{O}(y), \text{ or } \\ t_{O}(x) = t_{O}(y) = n \text{ and } x <_{O_{n}} y. \end{array}\right. \end{align}\] The induced total order \(<_{X}\) on \(X\) is then given as follows: for any \(x, y \in X\), \[\begin{align} x <_{X} y \Longleftrightarrow \left\{\begin{array}{l} t_{X}(x) < t_{X}(y), \text{ or } \\ t_{X}(x) = t_{X}(y) = n \text{ and } \left\{\begin{array}{l} C(x) <_{O_{n}} C(y), \text{ or }\\ C(x) = C(y) = z \in O_{+} \text{ and } x <_{C_{z}} y. \end{array}\right. \end{array}\right. \end{align}\] It is straightforward to verify that \((O, <_{O})\) forms a well-ordered set. By Lemma 6, the sets \((X, <_{X})\), \((\langle O \rangle, <_{\langle O \rangle})\), and \((\langle X \rangle, <_{\langle X \rangle})\) are well-ordered. Moreover, the lifted comultiplication \(\Delta\) is skew-triangular by construction. ◻

Proof. By induction on the word length, it suffices to consider the base case where \(\omega = y_{b} \in X_{+}\) for some \(y \in O_{+}\) and \(b \in O_{0}^{*}\). By utilizing the canonical algebra projection \(\pi\) constructed in Lemma 26, we obtain: \[\begin{align} \pi(ay_{b}) &= \pi(a) \pi(y_{b}) \\ & = \sum \pi(a) \pi(b)_{(1)} \pi(y) S_{H}(\pi(b)_{(2)}) \\ & = \sum \pi(a)_{(1)} \pi(b)_{(1)} \pi(y) S_{H}( \pi(b)_{(2)} ) S_{H}( \pi(a)_{(2)(1)} ) \pi(a)_{(2)(2)} \\ & = \sum \pi(a)_{(1)(1)} \pi(b)_{(1)} \pi(y) S_{H}( \pi(b)_{(2)} ) S_{H}( \pi(a)_{(1)(2)} ) \pi(a)_{(2)}. \end{align}\] By virtue of the comultiplication \(\Delta\) defined in Lemma 26, there exist elements \(a', a'' \in X_{0}^{*}\) such that \[\begin{align} \Delta_{H}(\pi(a)) &= (\pi \otimes \pi)(\Delta(a)) = \sum \pi(a') \otimes \pi(a''). \end{align}\] Substituting this expression into the identity for \(\pi(ay_{b})\) yields: \[\begin{align} \pi(ay_{b}) & = \sum \pi(a')_{(1)} \pi(b)_{(1)} \pi(y) S_{H}(\pi(b)_{(2)}) S_{H}(\pi(a')_{(2)}) \pi(a'') \\ & = \sum \pi(a'b)_{(1)} \pi(y) S_{H}(\pi( a'b)_{(2)} ) \pi(a''). \end{align}\] From Remark 27, it follows that each product \(a' b\) can be expressed modulo \(I\) as \(a' b \in \sum k_{a_{1}} a_{1}\) for some \(a_{1} \in X_{0}^{*}\) and \(k_{a_{1}} \in \mathbb{k}\). Setting \(a_{2} := a''\), we obtain: \[\begin{align} \pi(ay_{b}) = \sum k_{a_{1}} \pi(a_{1})_{(1)} \pi(y) S_{H}(\pi( a_{1})_{(2)} ) \pi(a_{2}) = \sum k_{a_{1}} \pi(y_{a_{1}}) \pi(a_{2}). \end{align}\] This equality implies that \(ay_{b} \in \sum k_{a_{1}} y_{a_{1}} a_{2} + I\). Furthermore, we observe that \(C(y_{a_{1}}a_{2}) <_{\langle O \rangle} C(ay_{b})\) holds because \[C(y_{a_{1}}a_{2}) \le_{\langle O \rangle} C( (y_{a_{1}}a_{2})_{L} ) = C(y_{a_{1}}) = y <_{\langle O \rangle} ay = C(ay_{b}).\] By definition of the reduction order on \(\langle X \rangle\), this inequality ensures that \(y_{a_{1}}a_{2} <_{\langle X \rangle} ay_{b}\). Consequently, the leading word of the corresponding relation in \(I\) is \(ay_{b}\), proving that \(ay_{b}\) is \(I\)-reducible. ◻

Proof. We proceed by induction on the length of the word \(\omega\). For \(|\omega| = 1\), the assertion follows immediately from the definition of the skew-triangular comultiplication on \(X\). Now, assume \(|\omega| \ge 2\) and decompose the word as \(\omega = x\alpha\) for some letter \(x \in X\). Since \(r(C(\alpha)) = r(C(\omega)) = C(m_{\omega}) = C(m_{\alpha})\), Lemma 15 (c) implies that \(\alpha_{L} = \alpha\).

Consequently, by the induction hypothesis (since \(|\alpha| < |\omega|\)), there exist elements \(x'_{+} \in X_{+}^{*}\), \(x'_{0} \in X_{0}^{*}\), \(\alpha'_{+} \in \langle X_{+} \rangle\), and \(\alpha'_{0} \in \langle X_{0} \rangle\) (setting \(\alpha' := \alpha'_{+}\alpha'_{0}\)), along with polynomials \(x'', \alpha'' \in \mathbb{k}\langle X \rangle^{+}\), such that \[\begin{align} \Delta(w) = & \Delta(x)\Delta(\alpha) \\ =& \left(x \otimes 1 + 1 \otimes x + \sum x'_{+}x'_{0} \otimes x''\right)\left(\alpha \otimes 1 + 1 \otimes \alpha + \sum \alpha' \otimes \alpha''\right)\\ =& x\alpha \otimes 1 + x \otimes \alpha + \sum x\alpha' \otimes \alpha''\\ & + \alpha \otimes x + 1 \otimes x\alpha + \sum \alpha' \otimes x\alpha''\\ & + \sum x'_{+}x'_{0} \alpha \otimes x'' + \sum x'_{+}x'_{0} \otimes x'' \alpha + \sum \sum x'_{+}x'_{0} \alpha' \otimes x''\alpha'', \end{align}\] where \(x'_{+}x'_{0} \neq 1\), \(\alpha' \neq 1\), \(C(x'_{+}) <_{O} C(x)\), and \(C(\alpha') \le_{\langle O \rangle} C((\alpha')_{L}) = C((\alpha'_{+})_{L}) <_{\langle O \rangle} C(\alpha)\).

By Lemma 28, there exist elements \(x'_{01}, x'_{02} \in X_{0}^{*}\), scalars \(k_{x_{01}'} \in \mathbb{k}\), and non-empty prime words \(\alpha_{x'_{01}} \in \langle X_{+} \rangle\) such that \[\begin{align} x_{0}' \alpha \in \sum k_{x_{01}'} \alpha_{x'_{01}} x'_{02} + I, \qquad \text{with } C(\alpha_{x'_{01}}) = C(\alpha). \end{align}\] Similarly, for each term \(\alpha'\) satisfying \(\alpha'_{+} \neq 1\), there exist scalars \(\ell_{x'_{01}} \in \mathbb{k}\) and non-empty prime words \((\alpha'_{+})_{x'_{01}} \in \langle X_{+} \rangle\) such that \[\begin{align} x_{0}' \alpha'_{+} \in \sum \ell_{x'_{01}}(\alpha'_{+})_{x'_{01}} x'_{02} + I, \qquad \text{with } C((\alpha'_{+})_{x'_{01}}) = C(\alpha'_{+}). \end{align}\] Substituting these relations back into the expression for \(\Delta(\omega)\), we obtain the following containment modulo \(I \otimes \mathbb{k}\langle X \rangle\): \[\begin{align} \Delta(w) \in \; & x \alpha \otimes 1 + x \otimes \alpha + \sum x\alpha' \otimes \alpha'' + \alpha \otimes x + 1 \otimes x\alpha + \sum \alpha' \otimes x\alpha''\\ & + \sum \sum k_{x_{01}'} x'_{+} \alpha_{x'_{01}} x'_{02} \otimes x'' + \sum x'_{+}x'_{0} \otimes x'' \alpha + \sum \sum_{\alpha'_{+} = 1} x'_{+} x'_{0} \alpha'_{0} \otimes x''\alpha'' \\ & + \sum \sum_{\alpha'_{+} \neq 1} \sum \ell_{x_{01}'} x'_{+}(\alpha'_{+})_{x'_{01}} x'_{02} \alpha'_{0} \otimes x''\alpha'' + I \otimes \mathbb{k}\langle X \rangle. \end{align}\]

By virtue of Lemma 16, it suffices to verify that every left-tensor word \(\omega'\) appearing in the sums above satisfies the condition \(C((\omega')_{L}) <_{\langle O \rangle} C(\omega)\). Since \(\omega\) is assumed to be a prime word, Lemma 16 ensures that \(C(\alpha) <_{\langle O \rangle} C(\omega)\) and \(C(x) <_{\langle O \rangle} C(\omega)\). We now check this condition case by case using Lemmas 9 and 16:

• Case \(\omega' = x\): We have \(C(x_{L}) = C(x) <_{\langle O \rangle} C(\omega)\).

• Case \(\omega' = x\alpha'\): Here, \(C((x\alpha')_{L}) = (C(x)C(\alpha'))_{L}\).

  • If \(m_{C(x)} >_{O} m_{C(\alpha')}\), then \((C(x)C(\alpha'))_{L} = C(x) <_{\langle O \rangle} C(\omega)\).

  • If \(m_{C(x)} \le_{O} m_{C(\alpha')}\), then \((C(x)C(\alpha'))_{L} = C(x) C(\alpha')_{L} <_{\langle O \rangle} C(x) C(\alpha) = C(\omega)\).

• Case \(\omega' = \alpha\): We have \(C(\alpha_{L}) = C(\alpha) <_{\langle O \rangle} C(\omega)\).

• Case \(\omega' = \alpha'\): We have \(C((\alpha')_{L}) <_{\langle O \rangle} C(\alpha) <_{\langle O \rangle} C(\omega)\).

• Case \(\omega' = x'_{+} \alpha_{x'_{01}} x'_{02}\): Here, \(C((x'_{+} \alpha_{x'_{01}} x'_{02})_{L}) = (C(x'_{+})C(\alpha))_{L}\).

  • If \(m_{C(x'_{+})} >_{O} m_{C(\alpha)}\), then \((C(x'_{+})C(\alpha))_{L} = C(x'_{+}) <_{\langle O \rangle} C(x) <_{\langle O \rangle} C(\omega)\).

  • If \(m_{C(x'_{+})} \le_{O} m_{C(\alpha)}\), then \((C(x'_{+})C(\alpha))_{L} = C(x'_{+}) C(\alpha)_{L} = C(x'_{+}) C(\alpha) <_{\langle O \rangle} C(x) C(\alpha) = C(\omega)\).

• Case \(\omega' = x'_{+}x'_{0}\): We have \(C((x'_{+}x'_{0})_{L}) = C(x'_{+}) <_{\langle O \rangle} C(x) <_{\langle O \rangle} C(\omega)\).

• Case \(\omega' = x'_{+}x'_{0} \alpha'_{0}\): We have \(C((x'_{+}x'_{0} \alpha'_{0})_{L}) = C(x'_{+}) <_{\langle O \rangle} C(x) <_{\langle O \rangle} C(\omega)\).

• Case \(\omega' = x'_{+}(\alpha'_{+})_{x'_{01}} x'_{02} \alpha'_{0}\) with \(\alpha_{+}' \neq 1\): Here, \(C((x'_{+}(\alpha'_{+})_{x'_{01}} x'_{02} \alpha'_{0})_{L}) = (C(x'_{+}) C(\alpha'_{+}))_{L}\).

  • If \(m_{C(x'_{+})} >_{O} m_{C(\alpha'_{+})}\), then \((C(x'_{+})C(\alpha'_{+}))_{L} = C(x'_{+}) <_{\langle O \rangle} C(x) <_{\langle O \rangle} C(\omega)\).

  • If \(m_{C(x'_{+})} \le_{O} m_{C(\alpha'_{+})}\), then \((C(x'_{+})C(\alpha'_{+}))_{L} = C(x'_{+}) C(\alpha'_{+})_{L} <_{\langle O \rangle} C(x'_{+})C(\alpha) <_{\langle O \rangle} C(x)C(\alpha) = C(\omega)\).

Combining all cases, it follows that every resulting term satisfies \(C(\omega') \le_{\langle O \rangle} C((\omega')_{L}) <_{\langle O \rangle} C(\omega)\), and consequently \(\omega' \le_{\langle X \rangle} (\omega')_{L} <_{\langle X \rangle} \omega\), completing the proof. ◻

Proof. Since \(\omega_{L}\) is a non-empty prime word in \(\langle X_{+} \rangle\), Part (a) follows directly from Lemmas 28 and 29. For Part (b), expanding the product via the algebra homomorphism property yields: \[\begin{align} \Delta(\omega) = & \Delta(\omega_{L}) \Delta(\omega_{R}) \\ \in & \left(\omega_{L} \otimes 1 + 1 \otimes \omega_{L} + \sum (\omega_{L})' \otimes (\omega_{L})''\right)\left(\omega_{R} \otimes [1-r_{\omega_{R}}] + 1 \otimes \omega_{R} + \sum (\omega_{R})' \otimes (\omega_{R})''\right) \\ & + I \otimes \mathbb{k}\langle X \rangle\\ \subseteq & \;\omega_{L}\omega_{R} \otimes [1-r_{\omega_{R}}] + \omega_{L} \otimes \omega_{R} + \sum \omega_{L} (\omega_{R})' \otimes (\omega_{R})'' + I \otimes \mathbb{k}\langle X \rangle\\ & + \omega_{R} \otimes \omega_{L} [1-r_{\omega_{R}}] + 1 \otimes \omega_{L} \omega_{R} + \sum (\omega_{R})' \otimes \omega_{L} (\omega_{R})'' \\ & + \sum (\omega_{L})' \omega_{R} \otimes (\omega_{L})'' [1-r_{\omega_{R}}] + \sum (\omega_{L})' \otimes (\omega_{L})'' \omega_{R} + \sum \sum (\omega_{L})' (\omega_{R})' \otimes \omega_{L}'' (\omega_{R})''. \end{align}\]

A straightforward application of Part (a) and Lemma 28 verifies that each left-tensor word \(\omega'\) arising in the expansion above satisfies the stated structural constraints. ◻

Proof. Let us define the index set of the first summation and its maximal component degree by \[\begin{align} W_{1} := \{f \mid p_{\beta} \in I \text{ and } k_{f} \neq 0\}, \qquad d_{1} := \max \{\deg(f) \mid f \in W_{1}\}. \end{align}\] If \(d_{1} = 0\), the assertion holds immediately. Assume therefore that \(d_{1} \ge 1\). Recall from Lemma 30 that there exist non-empty words \(\beta', f', g' \in \langle X_{+} \rangle\langle X_{0} \rangle\) and polynomials \(\beta'', f'', g'' \in \mathbb{k}\langle X \rangle^{+}\) such that applying the comultiplication yields: \[\begin{align} & \Delta\left(\beta + \sum_{f \in W_{1}} k_{f} \beta f + \sum_{g} k_{g} g\right) \\ \in & \;\beta \otimes 1 + 1 \otimes \beta + \sum \beta' \otimes \beta'' \\ & + \sum_{f \in W_{1}} k_{f} \left(\beta \otimes 1 + 1 \otimes \beta + \sum \beta' \otimes \beta''\right)\left(f \otimes [1-r_{f}] + 1 \otimes f + \sum f' \otimes f''\right) \\ & + \sum k_{g} \left(g \otimes [1-r_{g}] + 1 \otimes g + \sum g' \otimes g''\right) + I \otimes \mathbb{k}\langle X \rangle\\ \subseteq & \;\beta \otimes 1 + 1 \otimes \beta + \sum \beta' \otimes \beta'' \\ & + \sum_{f \in W_{1}} k_{f} \Big(\beta f \otimes [1-r_{f}] + \beta \otimes f + \sum \beta f' \otimes f'' \\ & + f \otimes \beta [1-r_{f}] + 1 \otimes \beta f + \sum f' \otimes \beta f'' \\ & + \sum \beta' f \otimes \beta'' [1-r_{f}] + \sum \beta' \otimes \beta'' f + \sum \sum \beta' f' \otimes \beta'' f''\Big) \\ & + \sum k_{g} \left(g \otimes [1-r_{g}] + 1 \otimes g + \sum g' \otimes g''\right) + I \otimes \mathbb{k}\langle X \rangle, \end{align}\] where \(C((\beta')_{L}) <_{\langle O \rangle} C(\beta)\), \(C(m_{f'}) \le_{O} C(m_{f}) <_{O} C(m_{\beta})\), and \((g')_{L} \le_{\langle X \rangle} g_{L} <_{\langle X \rangle} \beta\).

Let \(p_{1} := 1 + \sum_{f\in W_{1}} k_{f} f\). Fix an element \(f_{0} \in W_{1}\), and partition the set \(W_{1}\) into \[\begin{align} W_{f_{0}} := \{f \in W_{1} \mid [1-r_{f}] \in [1-r_{f_{0}}] + I\}, \qquad W_{1}' := W_{1} \setminus W_{f_{0}}. \end{align}\] We now proceed by analyzing the following two cases.

Case 1: Suppose \([1-r_{f_{0}}] \in \mathbb{k}p_{1} + I\). By applying the canonical augmentation map \(\epsilon\), we deduce that \(p_{1} \in [1-r_{f_{0}}] + I\). Substituting this relation back into our initial hypothesis directly yields \[\begin{align} \beta[1-r_{f_{0}}] + \sum_{g} k_{g} g \in I, \end{align}\] which provides the desired form.

Case 2: Suppose \([1-r_{f_{0}}] \notin \mathbb{k}p_{1} + I\). Then the sum \(\mathbb{k}p_{1} + I + \mathbb{k}[1-r_{f_{0}}]\) is direct. We can therefore decompose the free algebra as \(\mathbb{k}\langle X \rangle= \mathbb{k}p_{1} \oplus I \oplus \mathbb{k}[1-r_{f_{0}}] \oplus U\) for some choosing complement subspace \(U \subseteq \mathbb{k}\langle X \rangle\). We define a linear functional \(\phi: \mathbb{k}\langle X \rangle\rightarrow \mathbb{k}\) by setting: \[\begin{align} \phi(p_{1}) = 1, \quad \phi(I) = \phi(U) = 0, \quad \text{and} \quad \phi([1-r_{f_{0}}]) = 0. \end{align}\]

Applying the linear map \(\operatorname{id}\otimes \phi\) to the containment \(\Delta(p_{\beta}) \in I \otimes \mathbb{k}\langle X \rangle+ \mathbb{k}\langle X \rangle\otimes I\), we find that there exist non-empty words \(g_{1} \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{g_{1}} \in \mathbb{k}\) such that \[\begin{align} \label{formula:beta-betaf-g1} \beta + \sum_{f \in W_{1}'} k_{f} \beta f \phi([1-r_{f}]) + \sum_{f\in W_{1}}\sum k_{f} \beta f' \phi(f'') + \sum_{g_{1}} k_{g_{1}} g_{1} \in I, \qquad \text{with } (g_{1})_{L} <_{\langle X \rangle} \beta. \end{align}\tag{1}\] By construction, the strict inequality \(|W_{1}'| < |W_{1}|\) holds. By iterating this reduction process, we can systematically eliminate the second term in equation 1 . This ensures the existence of non-empty words \(w, g_{2} \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{w}, k_{g_{2}} \in \mathbb{k}\) such that \[\begin{align} \label{formula:beta-betaf-g2} \beta + \sum_{w} k_{w} \beta w + \sum_{g_{2}} k_{g_{2}} g_{2} \in I, \end{align}\tag{2}\] where \(\deg(w) < d_{1}\), \(C(m_{w}) <_{O} C(m_{\beta})\), and \((g_{2})_{L} <_{\langle X \rangle} \beta\).

Now, let us define the new supporting index set: \[\begin{align} W_{2} := \{w \mid w \text{ appears in } \eqref{formula:beta-betaf-g2} \text{ and } k_{w} \neq 0\}, \qquad d_{2} := \max \{\deg(w) \mid w \in W_{2}\}. \end{align}\] Since \(d_{2} < d_{1}\), a standard induction argument on the degree \(d_{1}\) allows us to transition from equation 2 to the final required relation ?? , completing the proof. ◻

Proof. Applying the canonical projection \(\pi\) to the given relation yields the following identity in \(H\): \[\begin{align} \overline{\beta} \left(\overline{1} + \sum_{q} k_{q} \overline{q}\right) + \sum_{h} k_{h} \overline{h} = 0. \end{align}\] Let us define the right-hand factor and its coproduct decomposition by \[\begin{align} v := \overline{1} + \sum_{q} k_{q} \overline{q}, \quad r := \operatorname{rank}(\Delta_{H}(v)), \quad \Delta_{H}(v) = \sum_{i=1}^{r} v'_{i} \otimes v''_{i}, \end{align}\] and denote the set of right tensor factors by \(V'' := \{v''_{1}, \ldots, v''_{r}\}\). By Lemma 30, there exist non-empty words \(\beta', h' \in \langle X_{+} \rangle\langle X_{0} \rangle\) and polynomials \(\beta'', h'' \in \mathbb{k}\langle X \rangle^{+}\) such that applying the coproduct \(\Delta_{H}\) gives: \[\begin{align} &\Delta_{H}\left(\overline{\beta} v + \sum_{h} k_{h} \overline{h}\right) \\ = & \;\left(\overline{\beta} \otimes \overline{1} + \overline{1} \otimes \overline{\beta} + \sum \overline{\beta'} \otimes \overline{\beta''}\right)\left(\sum_{i=1}^{r} v'_{i} \otimes v''_{i}\right) + \sum_{h} k_{h} \left(\overline{h} \otimes (\overline{1} - \overline{r_{h}}) + \sum \overline{1} \otimes \overline{h} + \sum \overline{h'} \otimes \overline{h''}\right)\\ = & \;\sum_{i=1}^{r} \overline{\beta} v'_{i} \otimes v''_{i} + \sum_{i=1}^{r} v'_{i} \otimes \overline{\beta} v''_{i} + \sum_{i=1}^{r} \overline{\beta'} v'_{i} \otimes \overline{\beta''} v''_{i} \\ & + \sum_{h} k_{h} \left(\overline{h} \otimes (\overline{1} - \overline{r_{h}}) + \sum \overline{1} \otimes \overline{h} + \sum \overline{h'} \otimes \overline{h''}\right), \end{align}\] where \((\beta')_{L} <_{\langle X \rangle} \beta\), and \((h')_{L} \le_{\langle X \rangle} h_{L} <_{\langle X \rangle} \beta\).

Recall from [18] that the set \(V''\) is linearly independent in \(H\). We can therefore decompose the Hopf algebra as \(H = \left(\bigoplus_{i=1}^{r} \mathbb{k}v''_{i}\right) \oplus U\) for a choice of complement subspace \(U \subseteq H\). For each \(v''_{i} \in V''\), we define a linear functional \(\phi_{v''_{i}} : H \rightarrow \mathbb{k}\) by setting: \[\begin{align} \phi_{v''_{i}} (v''_{j}) = \delta_{i, j} \quad \text{for } v''_{j} \in V'', \qquad \text{and} \qquad \phi_{v''_{i}}(U) = 0. \end{align}\] Applying the linear map \(\operatorname{id}\otimes \phi_{v''_{i}}\) to the identity \(\Delta_{H}\left( \overline{\beta} v + \sum k_{h} \overline{h}\right) = 0\), we find that there exist elements \(\eta \in \langle X_{+} \rangle^{+}\), \(u, y \in H_{[0]}\), and scalars \(k_{\overline{\eta} u}, k_{y} \in \mathbb{k}\) such that \[\begin{align} \overline{\beta} v'_{i} + \sum k_{\overline{\eta} u} \overline{\eta} u + \sum k_{y} y = 0, \qquad \text{with } \eta_{L} <_{\langle X \rangle} \beta. \end{align}\] Right-multiplying this equation by \(S_{H}(v''_{i})\) and summing over all \(i\) yields: \[\begin{align} \sum_{i=1}^{r} \overline{\beta} v'_{i} S_{H}(v''_{i}) + \sum_{i=1}^{r} \sum k_{\overline{\eta} u} \overline{\eta} u S_{H}(v''_{i}) + \sum_{i=1}^{r} \sum k_{y} y S_{H}(v''_{i}) = 0. \end{align}\] Since \(\epsilon_{H}(v) = 1\), the antipode property guarantees that \(\sum_{i=1}^{r} v'_{i}S_{H}(v''_{i}) = 1\). The identity simplifies to: \[\begin{align} \overline{\beta} + \sum_{i=1}^{r} \sum k_{\overline{\eta} u} \overline{\eta} u S_{H}(v''_{i}) + \sum_{i=1}^{r} \sum k_{y} y S_{H}(v''_{i}) = 0. \end{align}\] Lifting this back to the free algebra and applying the augmentation map \(\epsilon\), we deduce the existence of words \(z_{1} \in \langle X_{0} \rangle\), \(z_{2} \in \langle X_{0} \rangle^{+}\), and scalars \(k_{\eta z_{1}}, k_{z_{2}} \in \mathbb{k}\) such that \[\begin{align} \beta + \sum k_{\eta z_{1}} \eta z_{1} + \sum k_{z_{2}} z_{2} \in I, \end{align}\] where \((\eta z_{1})_{L} = \eta_{L} <_{\langle X \rangle} \beta\), and \((z_{2})_{L} = z_{2} <_{\langle X \rangle} \beta\). Setting \(\mu\) to be the words appearing in the lower-order terms completes the proof. ◻

Proof. If \(x \in I\), the assertion holds vacuously. Assume therefore that \(x \notin I\). We proceed via case analysis based on the structural type of the letter \(x\).

Case 1: Suppose \(x \in O_{+}\). By Lemma 28 and application of the canonical augmentation map \(\epsilon\), there exist non-empty words \(\omega \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{\omega} \in \mathbb{k}\) such that \[\begin{align} x \in \sum k_{\omega} \omega + I, \qquad \text{with } \omega <_{\langle X \rangle} x, \end{align}\] or equivalently, by Corollary 8, \[\begin{align} x \in \sum k_{\omega} \omega + I, \qquad \text{with } C(r(\omega)) <_{O} x. \end{align}\] Let us define the support set and its associated prime word set by \[\begin{align} W := \{\omega \mid x \in \sum k_{\omega} \omega + I \text{ and } k_{\omega} \neq 0\}, \qquad W_{L} := \{\omega_{L} \mid \omega \in W\}. \end{align}\] Let \(\beta\) denote the greatest word of \(W_{L}\) with respect to the order \(<_{\langle X \rangle}\). We can thus isolate the leading structural components of \(x\) as \[\begin{align} x \in \sum_{\beta\alpha \in W} k_{\beta \alpha} \beta \alpha + \sum_{\substack{\omega \in W \\ \omega_{L} <_{\langle X \rangle} \beta}} k_{\omega} \omega + I, \qquad \text{where } C(r(\beta\alpha)), C(r(\omega)) <_{O} x. \end{align}\] Without loss of generality, we may assume that the leading sum satisfies \(\sum_{\beta \alpha \in W} k_{\beta \alpha} \beta \alpha \notin I\).

Since \((\langle X \rangle, <_{\langle X \rangle})\) is well-ordered by Lemma 6, we can proceed by induction on \(\beta\) with respect to \(<_{\langle X \rangle}\). If \(\beta <_{\langle X \rangle} x\), then Lemma 15 (e) implies that for every word \(\omega \in W\) and each constituent letter \(b\) of \(\omega\), the inequality \(b \le_{\langle X \rangle} \omega_{L} \le_{\langle X \rangle} \beta <_{\langle X \rangle} x\) holds. Lemma 7 then ensures that \(b <_{X} x\), completing this subcase.

Conversely, assume that \[\begin{align} \beta \ge_{\langle X \rangle} x \qquad (\text{which translates to } C(r(\beta)) \ge_{O} x \text{ by Corollary~\ref{cor:well-orders-compatibility-2}}). \end{align}\] This implies that \(\alpha \neq 1\) for every word \(\beta \alpha \in W\), because \(C(r(\beta\alpha)) <_{O} x\).

Let \(\ell\) denote the minimal non-empty word in \(\langle X_{+} \rangle\), which corresponds to the least letter in \(X_{+}\) under \(<_{X}\) via Lemma 11 (b). If \(\beta = \ell\), then \(x \in \sum k_{\ell\alpha} \ell\alpha + \sum k_{\omega} \omega + I\), where \(\alpha \in \langle X_{0} \rangle\) and \(\omega \in \langle X_{0} \rangle^{+}\). Since \(\ell \ge_{\langle X \rangle} x\) and \(x \in X_{+}\), it must be that \(x = \ell\), which gives \(\ell - \sum k_{\ell\alpha} \ell\alpha - \sum k_{\omega} \omega \in I\). Lemma 32 then guarantees the existence of non-empty words \(u \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{u} \in \mathbb{k}\) such that \[\begin{align} \ell \in \sum k_{u} u + I, \qquad \text{with } u_{L} <_{\langle X \rangle} \ell. \end{align}\] This forces \(u \in \langle X_{0} \rangle^{+}\), matching the claim.

Now, suppose \(\beta >_{\langle X \rangle} \ell\). By Lemma 30 and the comultiplication on \(X\), there exist elements \(x'_{+} \in X_{+}^{*}\), \(x'_{0} \in X_{0}^{*}\), non-empty words \((\beta\alpha)', \omega' \in \langle X_{+} \rangle\langle X_{0} \rangle\), and polynomials \(x'', (\beta\alpha)'', \omega'' \in \mathbb{k}\langle X \rangle^{+}\) such that \[\begin{align} & \Delta\left( \sum_{\beta\alpha \in W} k_{\beta\alpha} \beta\alpha + \sum_{\substack{\omega \in W \\ \omega_{L} <_{\langle X \rangle} \beta}} k_{\omega} \omega - x \right) \\ \in & \sum_{\beta\alpha \in W} k_{\beta\alpha} \left(\beta\alpha \otimes [1-r_{\alpha}] + \beta \otimes \alpha + 1 \otimes \beta\alpha + \sum (\beta\alpha)' \otimes (\beta\alpha)''\right) \\ & + \sum_{\substack{\omega \in W \\ \omega_{L} <_{\langle X \rangle} \beta}} k_{\omega}\left(\omega \otimes [1-r_{\omega}] + 1 \otimes \omega + \sum \omega' \otimes \omega''\right) \\ & - \left(x \otimes 1 + 1 \otimes x + \sum x'_{+}x'_{0} \otimes x''\right) + I \otimes \mathbb{k}\langle X \rangle, \end{align}\] where \(x'_{+}x'_{0} \neq 1\), \((\beta\alpha)' \neq 1\), \(C((x'_{+}x'_{0})_{L}) = C(x'_{+}) <_{O} C(x)\), \((\omega')_{L} \le_{\langle X \rangle} \omega_{L} <_{\langle X \rangle} \beta\), and either \(C(((\beta\alpha)')_{L}) <_{\langle O \rangle} C(\beta)\) or \((\beta\alpha)' = \beta c\) for some \(c \in \langle X \rangle^{+}\) satisfying \(C(m_{c}) <_{O} C(m_{\beta})\).

Note that the subspace \(\mathbb{k}(\sum_{\beta\alpha \in W} k_{\beta\alpha} \alpha) + I\) forms a direct sum because \(\sum_{\beta\alpha \in W} k_{\beta\alpha}\alpha \notin I\). Since \(\alpha \neq 1\) for all \(\beta\alpha \in W\), the application of the augmentation map \(\epsilon\) shows that the sum \(\mathbb{k}(\sum_{\beta\alpha \in W} k_{\beta\alpha} \alpha) + I + \mathbb{k}1\) is also direct. We can therefore choose a complement decomposition of the free algebra of the form \(\mathbb{k}\langle X \rangle= \mathbb{k}(\sum_{\beta\alpha \in W} k_{\beta\alpha} \alpha) \oplus I \oplus \mathbb{k}1 \oplus V\) for some subspace \(V \subseteq \mathbb{k}\langle X \rangle\).

Define a linear functional \(\varphi : \mathbb{k}\langle X \rangle\rightarrow \mathbb{k}\) by setting: \[\begin{align} \varphi\left(\sum_{\beta\alpha \in W} k_{\beta\alpha} \alpha\right) = 1, \quad \varphi(1) = 0, \quad \text{and} \quad \varphi(I) = \varphi(V) = 0. \end{align}\] Applying the linear map \(\operatorname{id}\otimes \varphi\) to the containment relation for the coproduct inside \(I \otimes \mathbb{k}\langle X \rangle+ \mathbb{k}\langle X \rangle\otimes I\), we find that there exist non-empty words \(f, g \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{f}, k_{g} \in \mathbb{k}\) satisfying \[\begin{align} \beta + \sum k_{f} \beta f + \sum k_{g} g \in I, \qquad \text{with } C(m_{f}) <_{O} C(m_{\beta}) \text{ and } g_{L} <_{\langle X \rangle} \beta. \end{align}\] By invoking Lemmas 31 and 32, there exist non-empty words \(\mu \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{\mu} \in \mathbb{k}\) such that \[\begin{align} \beta \in \sum k_{\mu} \mu + I, \qquad \text{with } \mu_{L} <_{\langle X \rangle} \beta. \end{align}\] We can now rewrite the element \(x\) modulo \(I\) as follows: \[\begin{align} x \in \sum_{\beta\alpha \in W} \sum_{\mu} k_{\beta \alpha} k_{\mu} \mu \alpha + \sum_{\substack{\omega \in W \\ \omega_{L} <_{\langle X \rangle} \beta}} k_{\omega} \omega + I, \end{align}\] where \(C(r( \mu \alpha)), C(r(\omega)) <_{O} x\), and \(\mu_{L} <_{\langle X \rangle} \beta\). Since \(\mu_{L} <_{\langle X \rangle} \beta\) and \(C(m_{\alpha}) <_{O} C(m_{\beta})\), it follows that \((\mu\alpha)_{L} <_{\langle X \rangle} \beta\). Applying Lemma 28, we establish that there exist non-empty words \(\nu \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{\nu} \in \mathbb{k}\) such that \[\begin{align} x \in \sum k_{\nu} \nu + I, \qquad \text{with } \nu_{L} <_{\langle X \rangle} \beta. \end{align}\] By our induction hypothesis on \(\beta\) (since \(\nu_{L} <_{\langle X \rangle} \beta\)), there exist letters \(x_{1}, \ldots, x_{n} \in X\) and scalars \(k_{x_{1}\cdots x_{n}} \in \mathbb{k}\) such that \[\begin{align} x \in \sum k_{x_{1}\cdots x_{n}} x_{1} \cdots x_{n} + I, \qquad \text{with } x_{1}, \ldots, x_{n} <_{X} x. \end{align}\]

Case 2: Suppose \(x_{a} \in X_{+} \setminus O_{+}\) with \(x \in O_{+}\) and \(a \in O_{0}\).

By Lemma 28 and the evaluation under \(\epsilon\), we have \(x_{a} \in \sum k_{\omega} \omega + I\) for some non-empty words \(\omega \in \langle X_{+} \rangle\langle X_{0} \rangle\) satisfying \(\omega <_{\langle X \rangle} x_{a}\) and scalars \(k_{\omega} \in \mathbb{k}\). By definition of the order \(<_{\langle X \rangle}\), there exist elements \(a_{1} \in O_{0}^{*}\) and \(a_{2}, \ldots, a_{s} \in O_{0}\) satisfying \(a_{1} <_{O} \cdots <_{O} a_{s} <_{O} a\) such that \[\begin{align} x_{a} \in \sum_{1 \le i \le s} k_{x_{a_{i}}} x_{a_{i}} + \sum k_{\omega} \omega + I, \qquad \text{with } \omega <_{\langle X \rangle} x. \end{align}\] An analogous reduction argument to Case 1 applies here by tracking the behavior of \(\Delta(x_{a} - \sum_{1 \le i \le s} k_{x_{a_{i}}} x_{a_{i}})\) instead of \(\Delta(x)\). It follows that there exist letters \(y_{1}, \ldots, y_{m} \in X\) and scalars \(k_{y_{1}\cdots y_{m}} \in \mathbb{k}\) such that \[\begin{align} x_{a} \in \sum_{1 \le i \le s} k_{x_{a_{i}}} x_{a_{i}} + \sum k_{y_{1}\cdots y_{m}} y_{1} \cdots y_{m} + I, \end{align}\] where \(y_{1}, \ldots, y_{m} <_{X} x \le_{X} x_{a_{1}} <_{X} \cdots <_{X} x_{a_{s}} <_{X} x_{a}\). This completes the proof. ◻

Proof. Assume to the contrary that there exist an \(I\)-reducible letter \(x \in O\) and a letter \(a \in O_{0}\) such that \(x_{a}\) is \(I\)-irreducible. Since \(x\) is assumed to be \(I\)-reducible, Corollary 8 guarantees the existence of non-empty words \(\omega \in \langle X_{+} \rangle\langle X_{0} \rangle\) and scalars \(k_{\omega} \in \mathbb{k}\) such that \[\begin{align} x \in \sum k_{\omega} \omega + I, \qquad \text{with } C(r(\omega)) <_{O} x. \end{align}\] By utilizing the linear assignment \(f_{X}\) constructed in Lemma 26, we can express \(x_{a}\) modulo \(I\) as \(x_{a} \in \sum a_{1} x a_{2} + I\) for some elements \(a_{1}, a_{2} \in X_{0}^{*}\). This yields a polynomial relation of the form \[\begin{align} p := x_{a} - \sum k_{\omega} a_{1} \omega a_{2} \in I. \end{align}\] We now evaluate the rightmost factors of the terms appearing in the summation:

• If \(a_{2} \neq 1\), then \(C(r(a_{1}\omega a_{2})) = a_{2} <_{O} x\).

• If \(a_{2} = 1\), then \(C(r(a_{1}\omega)) = C(r(\omega)) <_{O} x\).

In either case, Corollary 8 implies that \(a_{1} \omega a_{2} <_{\langle X \rangle} x <_{\langle X \rangle} x_{a}\). Consequently, the leading word of the polynomial satisfies \(\mathrm{LW}(p) = x_{a}\), which forces \(x_{a}\) to be \(I\)-reducible, yielding a contradiction.

We now establish assertion (b). From the validity of Part (a), the inclusion \(C(X_{I,+}) \subseteq O_{I,+}\) follows directly. Conversely, since \(O_{I,+} = C(O_{I,+}) \subseteq C(X_{I,+})\), we obtain the reverse containment. Hence, the equality \(C(X_{I,+}) = O_{I,+}\) holds. ◻

Proof. Since \((X, <_{X})\) is well-ordered, we proceed by induction on reducible letters of \(X_{+}\) with respect to the order \(<_{X}\). By Lemma 33, there exist constituent letters \(x_{1}, \ldots, x_{n} \in X\) and scalars \(k_{x_{1}\cdots x_{n}} \in \mathbb{k}\) such that \[\begin{align} x \in \sum k_{x_{1}\cdots x_{n}} x_{1} \cdots x_{n} + I, \qquad \text{with } x_{1}, \ldots, x_{n} <_{X} x. \end{align}\] Applying the induction hypothesis to the terms on the right-hand side (since \(x_{i} <_{X} x\) for all \(i\)), there exist replacement letters \(y_{1}, \ldots, y_{m} \in X_{I,+} \cup X_{0}\) and scalars \(k_{y_{1} \cdots y_{m}} \in \mathbb{k}\) yielding the representation \[\begin{align} x \in \sum k_{y_{1}\cdots y_{m}} y_{1} \cdots y_{m} + I, \qquad \text{with } y_{1}, \ldots, y_{m} <_{X} x. \end{align}\] Thus, every reducible letter in \(X_{+}\) can be expressed modulo \(I\) as a linear combination of products of letters in \(X_{I,+} \cup X_{0}\). Since irreducible letters in \(X_{+}\) trivially satisfy the representation, it follows that all letters in \(X_{+}\) share this property.

By referencing the structural map \(f_{X}\) defined in Lemma 26, every letter in \(X_{+}\) can be written modulo \(I\) as a linear combination of products of elements from \(C(X_{I,+}) \cup X_{0}\). By Lemma 34, this is precisely the set \(O_{I,+} \cup X_{0}\). Extending this result factor by factor, it follows that any non-empty word in \(\langle X \rangle\) can be expressed modulo \(I\) as a linear combination of products of letters from \(O_{I,+} \cup X_{0}\), completing the proof. ◻

Proof. Assertion (a) follows directly from Lemma 26 and Theorem 35.

For assertion (b), first assume that \(H\) is left Noetherian. By Lemma 2, the Hopf coradical \(H_{[0]}\) is a Hopf subalgebra of \(H\). Since \(H_{[0]}\) is assumed to be affine, we may select a finite generating set \(G_{[0]}\) for \(H_{[0]}\). By virtue of Part (a), there exists a generating set \(X\) of \(H\) satisfying the conditions of Lemma 26 such that \(\pi(O_{I,+}) \cup G_{[0]}\) generates \(H\) as an algebra. Since \(H_{[0]}\) is left Noetherian, Lemma 24 guarantees that the set \(\pi(O_{I,+}) \cup G_{[0]}\) is finite, which implies that \(H\) is affine.

Conversely, if \(H\) is right Noetherian, then its opposite-coopposite Hopf algebra \(H^{\mathrm{op},\mathrm{cop}}\) is left Noetherian and consequently affine by the preceding argument. Since affineness is preserved under passing to the opposite-coopposite structure, it follows that \(H\) is also affine. ◻

Proof. To demonstrate that \(H_{[0]}\) is affine, it is sufficient to show the equality \(H_{[0]} = \pi(\mathbb{k}\langle Y_{0} \rangle)\). The containment \(\pi(\mathbb{k}\langle Y_{0} \rangle) \subseteq H_{[0]}\) holds by definition. For the reverse containment, let \(C\) be a simple subcoalgebra of \(H\), and let \(M\) be a simple right \(C\)-subcomodule of \(C\). Choose a non-zero element \(z \in M\). We proceed via a case analysis on the counit evaluation of \(z\) to demonstrate that \(z \in \pi(\mathbb{k}\langle Y_{0} \rangle)\).

Case 1: Suppose \(\epsilon_{H}(z) = 0\). Let \(r := \mathrm{rank}(\Delta_{H}(z)) - 2\). By the standard properties of tensor rank and counitality, there exist elements \(z_{i}', z_{i}'', g, h \in M\) such that \[\begin{align} \Delta_{H}(z) = z \otimes g + h \otimes z + \sum_{i=1}^{r} z_{i}' \otimes z_{i}'', \end{align}\] where \(\epsilon_{H}(z_{i}') = \epsilon_{H}(z_{i}'') = 0\), and \(\epsilon_{H}(g) = \epsilon_{H}(h) = 1\).

Since \(\pi(X)\) forms a generating set of \(H\), there exist words \(\omega \in \langle Y \rangle\) and scalars \(k_{\omega} \in \mathbb{k}\) such that \(z = \pi\left(\sum k_{\omega} \omega\right)\). Let us define the support set and its corresponding prime word set by \[\begin{align} W := \left\{\omega \;\middle|\; z = \pi\left(\sum k_{\omega} \omega\right) \text{ and } k_{\omega} \neq 0\right\}, \qquad W_{L} := \{D(\omega_{L}) \mid \omega \in W\}. \end{align}\] Let \(D(\zeta)\) denote the greatest word of \(W_{L}\) with respect to the well-ordering \(<_{\langle O \rangle}\). We expand \(z\) in terms of this leading component as \[z = \pi \left(\sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} k_{\beta\alpha} \beta\alpha + \sum_{\substack{\omega \in W \\ D(\omega_{L}) <_{\langle O \rangle} D(\zeta)}} k_{\omega} \omega \right).\]

We proceed by induction on \(D(\zeta)\) with respect to \(<_{\langle O \rangle}\). Applying the coproduct \(\Delta_{H}\) yields: \[\Delta_{H}(z) = (\pi \otimes \pi) \circ \Delta \left(\sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} k_{\beta\alpha} \beta\alpha + \sum_{\substack{\omega \in W \\ D(\omega_{L}) <_{\langle O \rangle} D(\zeta)}} k_{\omega} \omega \right).\] If \(\deg(\zeta) = 0\), then it follows that \(\omega \in \langle Y_{0} \rangle\) for all \(\omega \in W\), which immediately yields \(z \in \pi(\mathbb{k}\langle Y_{0} \rangle)\). Now, assume \(\deg(\zeta) \ge 1\). By Lemmas 39 and 40, there exist words \(\beta', \alpha', \omega' \in \langle Y \rangle\) and homogeneous polynomials \(\beta'', \alpha'', \omega'' \in \mathbb{k}\langle Y \rangle\) such that \[\begin{align} \label{formula:mainthm4951} & \quad \;z \otimes g + h \otimes z + \sum_{i=1}^{r} z_{i}' \otimes z_{i}'' \\ &= (\pi \otimes \pi) \Bigg( \sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} k_{\beta\alpha} \bigg( \sum_{\substack{D(\beta')= D(\beta) \\ \deg(\beta'')=0}} \beta'\alpha' \otimes \beta''\alpha'' + \sum_{D(\beta') <_{\langle O \rangle} D(\beta)} \beta'\alpha' \otimes \beta''\alpha'' \bigg), \notag \\ & \quad + \sum_{\substack{\omega \in W \\ D(\omega_{L}) <_{\langle O \rangle} D(\zeta)}} \sum k_{\omega} \omega' \otimes \omega'' \Bigg), \notag \end{align}\tag{3}\] where \(D((\omega')_{L}) \le_{\langle O \rangle} D(\omega_{L}) <_{\langle O \rangle} D(\zeta)\), \(D(m_{\alpha'}) \le_{O} D(m_{\alpha}) <_{O} D(m_{\zeta})\), and for each summand \(\alpha' \otimes \alpha''\) in \(\Delta(\alpha)\), \(\deg(\alpha') \le \deg(\alpha)\), with strict inequality precisely whenever \(\deg(\alpha'') > 0\).

Let us define the maximal component degree \(d_{1}\) and the right component subspace \(U\) by \[\begin{align} d_{1} &:= \max\{\deg(\alpha) \mid \beta\alpha \in W\}, \\ U &:= \sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} \sum_{\substack{D(\beta')= D(\beta) \\ \deg(\beta'')=0}} \mathbb{k}\pi(\beta''\alpha''). \end{align}\] We now branch into two distinct structural sub-cases:

Case 1.1: Suppose \(d_{1} = 0\). This implies that \(\alpha, \alpha' \in \langle Y_{0} \rangle\) and \(\alpha'' \in \mathbb{k}\langle Y_{0} \rangle\) for all elements. We subdivide this condition as follows:

Case 1.1.1: Suppose \(\mathbb{k}z \cap U \neq 0\). Since \(\deg(\beta'') = 0\) for all \(\beta''\) appearing in \(U\), we have \(\beta'' \in \mathbb{k}\langle Y_{0} \rangle\), forcing \(z \in \pi(\mathbb{k}\langle Y_{0} \rangle)\).

Case 1.1.2: Suppose \(\mathbb{k}z \cap U = 0\). The sum \(\mathbb{k}z + U\) is therefore direct. We can decompose the entire Hopf algebra as \(H = \mathbb{k}z \oplus U \oplus R\) for a choice of complement subspace \(R \subseteq H\). Define a linear functional \(\psi: H \rightarrow \mathbb{k}\) by setting: \[\begin{align} \psi(z) = 1, \quad \psi(U) = 0, \text{ and } \psi(R) = 0. \end{align}\] Applying the linear map \(\operatorname{id}\otimes \psi\) to equation 3 , we find that there exist a non-zero element \(w \in M\), words \(\mu \in \langle Y \rangle\), and scalars \(k_{\mu} \in \mathbb{k}\) such that \(w = \pi \left(\sum k_{\mu} \mu \right)\) with \(D(\mu_{L}) <_{\langle O \rangle} D(\beta) = D(\zeta)\).

Since \(M\) is a simple left \(C^{*}\)-module of \(C\), there exists a linear form \(c^{*} \in C^{*}\) such that \(z = (\operatorname{id}\otimes c^{*})(\Delta_{H}(w))\). By Lemmas 39 and 40, there exist words \(\nu \in \langle Y \rangle\) and scalars \(k_{\nu} \in \mathbb{k}\) satisfying \(z = \pi\left(\sum k_{\nu} \nu\right)\) with \(D(\nu_{L}) <_{\langle O \rangle} D(\zeta)\). By applying our induction hypothesis (since \(D(\nu_{L}) <_{\langle O \rangle} D(\zeta)\)), the claim follows.

Case 1.2: Suppose \(d_{1} \ge 1\). Let us define the restricted target subspace \[\begin{align} V := \sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} \sum_{\substack{D(\beta')= D(\beta) \\ \deg(\beta''\alpha'')=0}} \mathbb{k}\pi(\beta''\alpha''). \end{align}\]

Case 1.2.1: If \(\mathbb{k}z \cap V \neq 0\), an argument analogous to Case 1.1.1 applies directly.

Case 1.2.2: If \(\mathbb{k}z \cap V = 0\), the sum \(\mathbb{k}z + V\) is direct. We express the space as \(H = \mathbb{k}z \oplus V \oplus T\) for a choice of complement subspace \(T \subseteq H\). Define a linear functional \(\varphi: H \rightarrow \mathbb{k}\) by setting: \[\begin{align} \varphi(z) = 1, \quad \varphi(V) = 0, \text{ and } \varphi(T) = 0. \end{align}\] Applying \(\operatorname{id}\otimes \varphi\) to equation 3 , we obtain a non-zero element \(v \in M\), words \(\tau \in \langle Y \rangle\), and scalars \(k_{\beta'\alpha'}', k_{\tau}' \in \mathbb{k}\) such that \[\begin{align} \label{formula:mainthm49511} v = \pi \left( \sum_{\substack{\beta\alpha \in W \\ D(\beta) = D(\zeta)}} \sum_{\substack{D(\beta')= D(\beta) \\ \deg(\beta'')=0 \\ \deg(\alpha'')>0}} k_{\beta'\alpha'}' \beta' \alpha' + \sum k_{\tau}' \tau \right), \end{align}\tag{4}\] where \(D(\tau_{L}) <_{\langle O \rangle} D(\zeta)\), and \(\deg(\alpha') < d_{1}\) (since \(\deg(\alpha'') > 0\)).

Since \(M\) is a simple left \(C^{*}\)-module, it follows from Lemmas 39 and 40 that there exist words \(\eta, f, \gamma \in \langle Y \rangle\) and scalars \(k_{\eta f}'', k_{\gamma}'' \in \mathbb{k}\) such that \[\begin{align} z = \pi\left(\sum k_{\eta f}'' \eta f + \sum k_{\gamma}'' \gamma \right), \end{align}\] where \(D(\eta) = D(\zeta)\), \(D(m_{f}) <_{O} D(m_{\zeta})\), \(D(\gamma_{L}) <_{\langle O \rangle} D(\zeta)\), and \(\deg(f) < d_{1}\).

Let us denote \(d_{2} := \max\{\deg(f) \mid z = \pi\left(\sum k_{\eta f}'' \eta f + \sum k_{\gamma}'' \gamma \right) \}\). Note that \(d_{2} < d_{1}\) holds strictly. Iterating this reduction process down the degree filtration yields elements \(q \in \langle Y_{0} \rangle\), \(\theta, \xi \in \langle Y \rangle\), and scalars \(k_{\theta q}''', k_{\xi}''' \in \mathbb{k}\) such that \[\begin{align} z = \pi\left(\sum k_{\theta q}''' \theta q + \sum k_{\xi}''' \xi \right), \end{align}\] where \(D(\theta) = D (\zeta)\), \(D(m_{q}) <_{O} D(m_{\zeta})\), and \(D(\xi_{L}) <_{\langle O \rangle} D(\zeta)\). The remaining verification follows precisely the same path as outlined in Case 1.1.

Case 2: Suppose \(\epsilon_{H}(z) = 1\) with \(z \neq 1\). Let \(r := \mathrm{rank}(\Delta_{H}(z)) - 1\). By tensor rank and counitality considerations, there exist elements \(z_{i}', z_{i}'' \in M \cap \ker \epsilon_H\) such that \[\begin{align} \Delta_{H}(z) = z \otimes z + \sum_{i=1}^{r} z_{i}' \otimes z_{i}''. \end{align}\] The proof concludes by applying the exact reduction machinery of Case 1 to \(z\) and adapting the sub-case analysis where necessary. ◻

Proof. The implication (a) \(\Rightarrow\) (b) follows from Theorem 41. The implication (b) \(\Rightarrow\) (c) follows immediately from Lemma 2. The implication (c) \(\Rightarrow\) (a) is guaranteed by Remark 37 and Theorem 36. Finally, the assertion that any Hopf subalgebra containing \(H_{(0)}\) is affine follows directly from the stabilization of ascending chains provided by Corollary 38. ◻

Proof. Assume without loss of generality that \(H\) is left Noetherian. By [29], \(H\) is faithfully flat over \(T\). It then follows from [30] that \(T\) is left Noetherian. Evidently, \(T\) inherits cosemisimplicity from \(H\). By [25], the antipode \(S_{H}\) is bijective and satisfies \(S_{H}^{2}(C) = C\) for any simple subcoalgebra \(C\) of \(T\). Hence, \(T\) is Noetherian and locally affine. Since \(T\) is faithfully flat over all its Hopf subalgebras, Lemma 45 implies that \(T\) is affine. An analogous argument applies if \(H\) is right Noetherian. ◻

Proof. By invoking Lemma 2 and [1], the Hopf coradical \(H_{[0]}\) forms a left or right Noetherian Hopf subalgebra of \(H\). Lemma 45 then guarantees that \(H_{[0]}\) is affine. It follows immediately from Theorem 43 that the entire Hopf algebra \(H\) is affine. ◻