Linear compactness and combinatorial bialgebras

We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose equivalence classes generate linearly compact bialgebras under shifted shuffling and deconcatenation. We also extend some of the theory of combinatorial Hopf algebras to bialgebras that are not connected or of finite graded dimension. Finally, we discuss several examples of quasi-symmetric functions, not necessarily of bounded degree, that may be constructed via terminal properties of combinatorial bialgebras.


Introduction
The graded dual W P of the Hopf algebra of word quasi-symmetric functions has a basis given by the set of packed words, i.e., finite sequences w = w 1 w 2 · · · w n with {w 1 , w 2 , . . . , w n } = {1, 2, . . . , m} for some m ≥ 0. The product for this Hopf algebra is a shifted shuffling operation, while the coproduct is a variant of deconcatenation; for the precise definitions, skip to Section 2.3.
A fruitful method of constructing Hopf algebras of interest in combinatorics is to choose an equivalence relation ∼ on packed words and then form the subspace K (∼) P ⊂ W P spanned by the sums over each ∼-equivalence class κ E := w∈E w. A long list of well-known Hopf algebras can be realized as a subalgebra of W P in this way: for example, the noncommutative symmetric functions NSym [12], the Poirier-Reutenauer algebra PR [41], the K-theoretic Poirier-Reutenauer algebra KPR [37], the small multi-Malvenuto-Reutenauer Hopf algebra mMR [24], the Loday-Ronco algebra [4,26], and the Baxter Hopf algebra [14]. Similar Hopf algebra constructions involving equivalences on (signed) words and permutations have been explored in [10,39,40,43], among other places.
The subspace K (∼) P ⊂ W P is not necessarily a sub-bialgebra, and one of the aims of this paper is to describe precisely when this occurs. The Hopf algebra W P is a quotient of a larger bialgebra W with a basis given by arbitrary words. We will also consider the problem of classifying the word relations that span sub-bialgebras K (∼) ⊂ W in a similar manner.
For homogeneous relations, versions of these problems have been studied in a few places previously, e.g., [14,19,35,42]. Less has been written about the cases when ∼ is allowed to relate words of different lengths. For inhomogeneous relations of this kind, various complications arise when one tries to interpret K (∼) P as an algebra or a coalgebra. To start, such relations may have equivalence classes with infinitely many elements, in which case K (∼) P contains infinite linear combinations of packed words so is not technically a subspace of W P . One can still try to evaluate the product and coproduct of W P on elements of K (∼) P when this happens. However, products may result in infinite linear combinations of the basis elements κ E , and even if these infinite sums are adjoined to K (∼) P , coproducts may have too many terms to belong to K (∼) Nevertheless, some interesting "Hopf algebras" that can be identified with K (∼) P when ∼ is inhomogeneous have appeared in the literature [16,24,36,37]. A secondary, expository goal of this paper is to describe explicitly the monoidal category containing such objects, which in general is not the usual category of bialgebras over a field. This point is often glossed over in the relevant combinatorial literature, though authors tend to indicate correctly that its resolution is topological in nature.
In detail, to make sense of "sub-bialgebras" of W P "spanned" by inhomogeneous word relations, one should first consider the larger vector spaceŴ P consisting of arbitrary (rather than just finite) linear combinations of packed words. This object is naturally viewed as a linearly compact topological space. The full subcategory of such spaces, within the category of all topological vector spaces, has a symmetric monoidal structure which leads to notions of linearly compact algebras, coalgebras, and bialgebras, of whichŴ P is an example. In this language, our original classification problem becomes the question: for which word relations ∼ is the subspaceK (∼) P , whose elements are the arbitrary linear combinations of the sums κ E , a linearly compact sub-bialgebra ofŴ P ?
After some preliminaries in Section 2, we review the main properties of linearly compact vector spaces in Section 3. This background material is semi-classical but perhaps not so widely known in combinatorics. Section 4 goes on to discuss some novel generalizations of the monoidal structures on W and W P . In Section 5, we answer the question in the previous paragraph. Our general results about word relations recover a number of specific constructions of (linearly compact) Hopf algebras and bialgebras; we discuss some relevant examples in Section 6.
One application of all this formalism is to extend Aguiar, Bergeron, and Sottile's theory of combinatorial Hopf algebras from [1]. Ignoring some technical details which will be clarified in Section 7, a combinatorial Hopf algebra over a field k is a Hopf algebra H with an algebra morphism ζ : H → k called the character. A morphism (H, ζ) → (H ′ , ζ ′ ) of combinatorial Hopf algebras is a Hopf algebra morphism φ : H → H ′ with ζ = ζ ′ • φ. The Hopf algebra of quasi-symmetric functions QSym with the homomorphism ζ QSym : QSym → k setting x 1 = 1 and x 2 = x 3 = · · · = 0 is a fundamental example.
It is shown in [1] that if (H, ζ) is a combinatorial Hopf algebra in which H is (1) graded, (2) connected, and (3) of finite graded dimension, then there is a unique morphism (H, ζ) → (QSym, ζ QSym ). This morphism supplies a uniform construction of many independent definitions of quasi-symmetric generating functions attached to Hopf algebras. In Section 7, we prove two extensions of this result. The first (see Theorem 7.4) removes assumptions (2) and (3), essentially just by reframing the character of H as an algebra morphism ζ : H → k[t]. The second (see Theorem 7.8) lifts all of the assumptions (1), (2), and (3), at the cost of introducing some topological conditions and replacing QSym by an appropriate completion.
These results are not unexpected; the authors mention in [1,Remark 4.2] that assumption (3) may be dropped, and note work in preparation where this will be proved. The relevant paper cited in [1,Remark 4.2] does not seem to have ever appeared in the literature, however. We hope that our exposition fills this gap.
In Section 8 we illustrate some more applications. We discuss several examples of families of symmetric and quasi-symmetric functions, not necessarily of bounded degree, that can be realized as the images of canonical morphisms from what we call (linearly compact) combinatorial bialgebras. For appropriate word relations, the spaceK (∼) P is an object of this type and is therefore equipped with a canonical morphism to a certain linearly compact "completion" of QSym. Our last results give a partial classification of the relations ∼ for which the image of this morphism consists entirely of symmetric functions. Definition 2.1. A monoid in C is a triple (A, ∇, ι) where A ∈ C is an object and ∇ : A • A → A and ι : I → A are morphisms (referred to as the product and unit) making these diagrams commute: A comonoid in C is a triple (A, ∆, ǫ) where A ∈ C is an object and ∆ : A → A • A and ǫ : A → I are morphisms (referred to as the coproduct and counit) making the diagrams (2.1), with ∇ and ι replaced by ∆ and ǫ and with the directions of all arrows reversed, commute.
Definition 2.3. A bimonoid in C is a tuple (A, ∇, ι, ∆, ǫ) where (A, ∇, ι) is a monoid, (A, ∆, ǫ) is a comonoid, the composition ǫ • ι is the identity morphism I → I, and these diagrams commute: A morphism of (bi, co) monoids is a morphism in C that commutes with the relevant (co)unit and (co)product morphisms. If A is a monoid then A•A is a monoid with product (∇•∇)•(id•β•id) and unit (ι • ι) • (I ∼ − → I • I). If A is a comonoid then A • A is naturally a comonoid in a similar way. The diagrams (2.2) express that the coproduct and counit of a bimonoid are monoid morphisms, and that the product and unit are comonoid morphisms.
We are exclusively interested in these definitions applied to a few related categories. Let k be a field and write Vec k for the usual category of k-vector spaces with linear maps as morphisms. This category is symmetric monoidal relative to the standard tensor product ⊗ = ⊗ k and braiding map x ⊗ y → y ⊗ x, with unit object k. Monoids, comonoids, and bimonoids in this category are the familiar notions of k-algebras, k-coalgebras, and k-bialgebras. In this context, the unit ι : k → A is completely determined by ι(1) ∈ A, which we refer to as the unit element.
Assume that C is k-linear so that the morphisms between any two fixed objects in C form a k-vector space. Let (H, ∇, ι, ∆, ǫ) be a bimonoid in C . The convolution product of two morphisms f, g : H → H is then f * g = ∇ • (f • g) • ∆ : H → H. The operation * is associative and makes the vector space of morphisms H → H into a k-algebra with unit element ι • ǫ, referred to as the convolution algebra of H. The bimonoid H is a Hopf monoid if the identity morphism id : H → H has a left and right inverse S : H → H in the convolution algebra. The morphism S is called the antipode of H; if it exists, then S is the unique morphism H → H such that Hopf monoids in Vec k are Hopf algebras.

Graded vector spaces
If I is a set and V i for i ∈ I is a k-vector space, then i∈I V i is the vector space of sums i∈I v i where v i ∈ V i for i ∈ I and v i = 0 for all but finitely many indices i ∈ I. We interpret the direct product i∈I V i as the vector space of arbitrary formal sums i∈I v i with v i ∈ V i . There is an obvious inclusion i∈I V i ⊂ i∈I V i which is equality if I is finite.
A vector space V is graded if it has a direct sum decomposition V = n∈N V n . A linear map φ : U → V between graded vector spaces is graded if it has the form φ = n∈N φ n where each φ n : U n → V n is linear. If U = n∈N U n and V = n∈N V n are direct products of vector spaces, then we also use the term graded to refer to the linear maps φ : U → V of the form φ = n∈N φ n where each φ n : U n → V n is linear.
for all n ∈ N, and ǫ(V n ) = 0 for n ∈ P. A bialgebra is graded if it is graded as both an algebra and a coalgebra. These notions correspond to (co, bi) monoids in the category GrVec k whose objects are graded k-vector spaces V = n∈N V n and whose morphisms are graded linear maps, in which the tensor product of objects U and V is the graded vector space The unit object in GrVec k is the field k, graded such that all elements have degree zero.

Word bialgebras
We review the definition of a particular graded bialgebra which will serve as a running example in later sections. Throughout, we use the term word to mean a finite sequence of positive integers. If w = w 1 w 2 · · · w n is a word with n letters and I = {i 1 < i 2 < · · · < i k } ⊂ [n] is a subset of indices, then we set w| I = w i 1 w i 2 · · · w i k . The shuffle product of two words u and v of length m and n is the formal linear combination of words is the unique (m + n)-letter word w with w| I = u and w| I c = v. Multiplicities may result in this expression; for example, 12 ¡ 21 = 2 · 1221 + 1212 + 2121 + 2 · 2112.
If w = w 1 w 2 · · · w m is a word with m > 0 letters, then we set max(w) = max{w 1 , w 2 , . . . , w m }. For the empty word ∅, we define max(∅) = 0. Let W n for n ∈ N be the set of pairs [w, n] with max(w) ≤ n and define W = n∈N W n . Let W n = kW n be the k-vector space with W n as a basis and define W = n∈N W n .
Denote the word formed by adding n ∈ N to each letter of w = w 1 w 2 · · · w m by w ↑ n = (w 1 + n)(w 2 + n) · · · (w m + n).
Given words w 1 , w 2 , . . . , w l with max(w i ) ≤ n and a 1 , a 2 , . . . , a l ∈ k, let for [w, n] ∈ W n with w = w 1 w 2 · · · w m . Finally write ι ¡ for the linear map k → W with ι ¡ (1) = [∅, 0]. We consider W to be a graded vector space in which [w, n] ∈ W n is homogeneous with degree ℓ(w), the length of the word w. The following is [32,Theorem 3.5]: is a graded bialgebra, but not a Hopf algebra.
A packed word (also called a surjective word [18], Fubini word [38], or initial word [37]) is a word w with w = fl(w). Define I P to be the subspace of W spanned by all differences [v, m] The following is [32, Proposition 3.7]: Proposition 2.5. The subspace I P is a homogeneous bi-ideal of (W, ∇ ¡ , ι ¡ , ∆ ⊙ , ǫ ⊙ ). The quotient bialgebra W P = W/I P is a graded Hopf algebra.
The Hopf algebra W P is the graded dual of the algebra of word quasi-symmetric functions WQSym [33,35]. Let W P be the set of all packed words. If [w, n] ∈ W and w is a word with m distinct letters then v = fl(w) is the unique packed word such that [w, n] + I P = [v, m] + I P . Identify v ∈ W P with the coset [v, m] + I P so that we can view W P as a basis for W P . The unit element of W P is then the empty packed word ∅, and the counit is the linear map ǫ ⊙ : W P → k with ǫ ⊙ (∅) = 1 and ǫ ⊙ (w) = 0 for all ∅ = w ∈ W P . For u, v, w ∈ W P with m = max(u) and n = ℓ(w), (2.5) The subspace of W P spanned by the words in W P that have no repeated letters is a Hopf subalgebra. This is the well-known Malvenuto-Poirier-Reutenauer Hopf algebra of permutations [3,29], sometimes also called the Hopf algebra of free quasi-symmetric functions FQSym [35].

Linearly compact spaces
Let U and V be k-vector spaces. Define U * to be the dual space of U , that is, the vector space of all k-linear maps λ : U → k. Given a linear map φ : U → V , define φ * to be the linear map V * → U * with φ * (λ) = λ • φ. This makes * into a contravariant functor Vec k → Vec k . We would like to be able to consider "sub-bialgebras" of W generated by certain infinite linear combinations of basis elements in W. Such linear combinations are not well-defined in W but are naturally interpreted as elements of W * . Therefore, we need a way of transferring the monoidal structures on the vector space W to its dual.
The full dual of an infinite-dimensional k-algebra is not naturally a k-coalgebra; see [9, §3.5]. On the other hand, neither the standard form of graded duality nor the more general notion of restricted duality (see [9, §3.5]) suffices for our application, since W does not have finite graded dimension and since the restricted dual will not permit infinite linear combinations.
The solution to these obstructions is to give the dual space a topology and consider monoidal structures in the category of topological vector spaces rather than Vec k . The topology in question is known as the linearly compact topology, whose properties we quickly review. Much of the background material in this section appears in [11,Chapter 1], so we omit some proofs.
A bilinear form ·, · : For example, the tautological form u, λ := λ(u) is a nondegenerate bilinear form U × U * → k. The bilinear form a, b := ab is likewise a nondegenerate pairing k × k → k.
Suppose ·, · : U × V → k is a nondegenerate bilinear form and {u i : i ∈ I} is a basis for U . For each i ∈ I, there exists a unique v i ∈ V with u j , v i = δ ij for all j ∈ I. As U = i∈I ku i , Lemma 3.1 implies that V = i∈I kv i . Thus each v ∈ V can be uniquely expressed as the (potentially infinite) sum v = i∈I u i , v v i . Following [11], we call {v i : i ∈ I} a pseudobasis for V ; this is sometimes also referred to as a continuous basis (e.g., in [36, §3]).
View each subspace kv i as a discrete topological space and give V = i∈I kv i the corresponding product topology; this is the linearly compact topology on V , also sometimes called the pseudocompact topology. This topology depends on the form ·, · but not on the choice of basis for U . Any finite intersection of sets of the form i∈I c i v i ∈ V : c j ∈ C for fixed choices of C ⊂ k and j ∈ I is open in the linearly compact topology, and every open subset of V can be expressed as a union of these intersections. In other words, a basis for the linearly compact topology consists of the sets for any finite list of indices i 1 , i 2 , . . . , i p ∈ I and any nonempty subsets C 1 , C 2 , . . . , C p ⊂ k. If V is finite-dimensional, then the linearly compact topology is discrete.

Definition 3.2.
A linearly compact k-vector space is a k-vector space V equipped with the linearly compact topology induced by a nondegenerate bilinear form U × V → k for some k-vector space U . Let Vec k denote the full subcategory of the category of topological k-vector spaces whose objects are linearly compact vector spaces.
As noted in [11], a topological vector space V belongs to Vec k if and only if its topology is Hausdorff and linear (i.e., the open affine subspaces form a basis) and any family of closed affine subspaces with the finite intersection property has nonempty intersection. The category Vec k is closed under arbitrary direct products and finite direct sums, and contains the category of finitedimensional vector spaces as a full subcategory.
A morphism between linearly compact vector spaces is a linear map that is continuous in the linearly compact topology. We can be more explicit about which linear maps are continuous. Suppose V, W ∈ Vec k have pseudobases {v i : i ∈ I} and {w j : j ∈ J}. Let ψ : V → W be a linear map and define ψ ij ∈ k to be the coefficient such that ψ(v i ) = j∈J ψ ij w j for all i ∈ I. Lemma 3.3. The map ψ : V → W is continuous in the linearly compact topology if and only if {i ∈ I : ψ ij = 0} is finite for each j ∈ J and ψ i∈I c i v i = j∈J i∈I c i ψ ij w j for any c i ∈ k.
In other words, ψ is continuous when i∈I c i ψ(v i ) is always defined and equal to ψ i∈I c i v i . It is an instructive exercise to work through the proof of this basic lemma.
Proof. If the given properties hold then the inverse image of i∈J c i w i ∈ W : c j ∈ C under ψ is a union of finite intersections of analogous sets in V and is therefore open. It follows in this case that the inverse image of any open subset of W under ψ is open, so ψ is continuous.
Conversely, assume ψ is continuous. Let j ∈ J. We first check that {i ∈ I : ψ ij = 0} is finite. Consider the open subset S = { k∈J c k w k ∈ W : c j = 0}. The inverse image ψ −1 (S) is open since ψ is continuous and nonempty since 0 ∈ S. Therefore ψ −1 (S) contains an open subset of the form (3.1). Let i 1 , i 2 , . . . , i p ∈ I be the finite list of indices corresponding to this subset. Then for any g ∈ ψ −1 (S) and i ∈ I \ {i 1 , i 2 , . . . , i p } we have g + v i ∈ ψ −1 (S), so ψ(g) ∈ S and ψ(g + v i ) ∈ S, whence by linearity ψ(v i ) = k∈J ψ ik w k ∈ S. But this says precisely that if i ∈ I \ {i 1 , i 2 , . . . , i p } then ψ ij = 0, so {i ∈ I : ψ ij = 0} is a subset of the finite set {i 1 , i 2 , . . . , i p }.
The map φ : V → W defined by φ i∈I c i v i = j∈J i∈I c i ψ ij w j is thus well-defined and linear, and also continuous by the first paragraph of the proof. Since ψ − φ is then linear and continuous, to deduce that ψ = φ, it suffices to show that the only continuous linear map Suppose we have nondegenerate bilinear forms ·, · i : Corollary 3.4. In the preceding setup, a linear map ψ : V 1 → V 2 is continuous in the linearly compact topology if and only if ψ = φ ⊥ for some linear map φ : The set of continuous linear maps V → W between linearly compact vector spaces is therefore a k-vector space. Let V ∨ be the vector space of continuous linear maps V → k for V ∈ Vec k . This vector space is sometimes called the continuous dual of V (for example, in [24, §7.4]).
Corollary 3.5. Suppose ·, · : U × V → k is a nondegenerate bilinear form. If {u i : i ∈ I} is a basis for U , then the functions u i , · : V → k for i ∈ I are a basis for V ∨ .
If ψ : V → W is a continuous linear map then ψ * : W * → V * restricts to a map W ∨ → V ∨ , which we denote ψ ∨ . The operation ∨ is then a contravariant functor Vec k → Vec k . The preceding corollary implies that U ∈ Vec k is naturally isomorphic to (U * ) ∨ as a vector space and that V ∈ Vec k is naturally isomorphic to (V ∨ ) * as a topological vector space. Thus, if V ∈ Vec k then the tautological pairing V ∨ × V → k is nondegenerate and the linearly compact topology induced by this form recovers the topology on V . We can summarize these observations as follows: Proposition 3.6. The functors * : Vec k → Vec k and ∨ : Vec k → Vec k are dualities of categories.
Define the completion of a k-vector space U with respect to a given basis {u i : i ∈ I} to be the vector spaceÛ = i∈I ku i with the product topology, where each subspace ku i is discrete. In other words,Û is the linearly compact k-vector space with {u i : i ∈ I} as a pseudobasis. Of course, if U is finite-dimensional then U =Û . The bilinear form ·, · : U × U → k with u i , u j = δ ij extends to a nondegenerate bilinear form U ×Û → k. The spaceÛ is distinguished from U * in having a fixed inclusion U ⊂Û . Relative to this inclusion, U is a dense subset ofÛ , which explains whyÛ is referred to as a completion.
The category Vec k has the following monoidal structure. For objects V, W, V ′ , W ′ ∈ Vec k and morphisms φ : V → V ′ and ψ : W → W ′ , define The object V⊗ W is a linearly compact vector space and the linear map φ⊗ ψ is continuous in the linearly compact topology. There is a canonical inclusion V ⊗ W ֒→ V⊗ W given by the linear map identifying v⊗w for v ∈ V and w ∈ W with the linear function that has λ⊗µ → λ(v)µ(w) for λ ∈ V ∨ and µ ∈ W ∨ . Relative to this inclusion, V ⊗ W is a dense subset of the linearly compact space V⊗ W , and for this reason one calls⊗ the completed tensor product. If V and W have pseudobases {v i : i ∈ I} and {w j : j ∈ J}, then the image of the set {v i ⊗ w j : This map uniquely extends to an isomorphismβ : V⊗ W → W⊗ V for all V, W ∈ Vec k . Recall that k is a linearly compact vector space with the discrete topology.
Proposition 3.7. The category Vec k is symmetric monoidal relative to the completed tensor product⊗, braiding mapβ, and unit object k.
Proof. Checking this proposition is a routine exercise from the axioms [2, Chapter 1]. One may simply transfer all arguments in the proof that Vec k is symmetric monoidal to Vec k by duality.
Since Vec k is symmetric monoidal, we have corresponding notions of (co, bi, Hopf) monoids in this category. We refer to monoids, comonoids, bimonoids, and Hopf monoids in Vec k respectively as linearly compact algebras, coalgebras, bialgebras, and Hopf algebras. A structure of this type consists explicitly of a linearly compact vector space V ∈ Vec k along with continuous linear maps V⊗ V → V , k → V , V → V⊗ V , and V → k satisfying the conditions in Section 2.1.
Alternatively, one can define linearly compact (co, bi, Hopf) algebras in Vec k entirely in terms of (co, bi, Hopf) algebras by duality. Let U ∈ Vec k and V ∈ Vec k and let ·, · : U × V → k be a nondegenerate bilinear form. Define u 1 ⊗ u 2 , v 1 ⊗ v 2 = u 1 , v 1 u 2 , v 2 for u i ∈ U and v i ∈ V and extend by continuity and linearity to define a nondegenerate bilinear form (U ⊗ U ) × (V⊗ V ) → k that is continuous in the second coordinate. Also let a, b = ab for a, b ∈ k. Now suppose ∇ : U ⊗ U → U , ι : k → U , ∆ : U → U ⊗ U , and ǫ : U → k are linear maps and ∇ : V⊗ V → V ,ι : k → V ,∆ : V → V⊗ V , andǫ : V → k are continuous linear maps such that for all u 1 , u 2 ∈ U , v ∈ V , and a ∈ k and for all u ∈ U , v 1 , v 2 ∈ V , and b ∈ k. Either map in each of the pairs (∇,∆), (ι,ǫ), (∆,∇) and (ǫ,ι) then uniquely determines the other. In this setup, (U, ∇, ι) is an algebra if and only if (V,∆,ǫ) is a linearly compact coalgebra; (U, ∆, ǫ) is a coalgebra if and only if (V,∇,ι) is a linearly compact algebra; and (U, ∇, ι, ∆, ǫ) is a bialgebra (respectively, Hopf algebra) if and only if (V,∇,ι,∆,ǫ) is a linearly compact bialgebra (respectively, Hopf algebra). In these cases, we say that the monoidal structure on V is the (algebraic) dual of the structure on U via the form ·, · . This perspective indicates how to give a linearly compact (co, bi, Hopf) algebra structure to the completed tensor product or direct sum of two linearly compact (co, bi, Hopf) algebras. For example, suppose U 1 and U 2 are algebras and V i is the linearly compact coalgebra dual to U i . Then U 1 ⊗ U 2 and U 1 ⊕ U 2 are both naturally algebras, and we can identify V 1⊗ V 2 with the dual of U 1 ⊗ U 2 and V 1 ⊕ V 2 with the dual of U 1 ⊕ U 2 in order to interpret both objects as linearly compact coalgebras. A similar statement holds if we assume each U i is a coalgebra, bialgebra, or Hopf algebra so that each V i is a linearly compact algebra, bialgebra, or Hopf algebra, respectively. The space k[x] is a graded Hopf algebra whose coproduct, counit, and antipode are the algebra morphisms with ∆(x) = 1 ⊗ x + x ⊗ 1, ǫ(x) = 0, and S(x) = −x. The space k[[x]] is a linearly compact Hopf algebra whose coproduct, counit, and antipode are the linearly compact algebra morphisms with the same formulas.
The Example 3.9. Any graded (co, bi, Hopf) algebra of finite graded dimension (that is, whose homogeneous components are each finite-dimensional) extends to a linearly compact (co, bi, Hopf) algebra. In detail, suppose V = n∈N V n is a graded k-vector space where each V n is finitedimensional. LetV = n∈N V n and give this space the product topology in which each subspace V n is discrete. ThenV is a linearly compact vector space and any graded linear map V ⊗ V → V or k → V or V → V ⊗ V or V → k extends uniquely to a continuous linear mapV⊗V →V or k →V orV →V⊗V orV → k, respectively. If V has the structure of a graded (bi, co, Hopf) algebra, then these extensions makeV into a linearly compact (bi, co, Hopf) algebra; the relevant structure onV is isomorphic to the algebraic dual of the graded dual of V .
Remark. Linearly compact (bi, co, Hopf) algebras have appeared in a few places previously in the literature, usually without being so named. For example, the "bialgebras"Γ andΛ in [6, §9] are linearly compact bialgebras. Likewise, the "Hopf algebras" mSym, mQSym, and mMR introduced in [24] and further studied in [36] are all linearly compact Hopf algebras.
Recall that W is the set of pairs [w, n] where n ∈ N and w is a word with letters in {1, 2, . . . , n}, and W = kW. DefineŴ to be the completion of W with respect to the basis W. For σ ∈Ŵ and [w, n] ∈ W, let σ(w, n) ∈ k denote the coefficient such that σ = [w,n]∈W σ(w, n)[w, n]. The associated nondegenerate bilinear form ·, · : W ×Ŵ → k is then for σ ∈ W and τ ∈Ŵ.
Define ∇ ⊙ :Ŵ⊗Ŵ →Ŵ to be the continuous linear map with and where w ∩ S denotes the subword of w formed by omitting all letters not in S. Define ǫ ¡ :Ŵ → k and ι ⊙ : k →Ŵ to be the continous linear maps with Proof. It is a straightforward exercise to check that (Ŵ, DefineŴ P to be the completion of the vector space of packed words W P with respect to the basis W P . The natural pairing W P ×Ŵ P → k givesŴ P the structure of a linearly compact Hopf algebra dual to (W P , ∇ ¡ , ι ¡ , ∆ ⊙ , ǫ ⊙ ), which one can realize as a sub-bialgebra of (Ŵ, ∇ ⊙ , ι ⊙ , ∆ ¡ , ǫ ¡ ). This object is not of much relevance to our discussion, however.
On the other hand, since W P has finite graded dimension when graded by word length, the maps ∇ ¡ , ι ¡ , ∆ ⊙ and ǫ ⊙ from (2.5) have continuous linear extensions to maps betweenŴ P , W P⊗ŴP , and k as appropriate, and the following holds in view of Example 3.9: LetŴ n be the completion of W n with respect to W n . Each subspace W n for n ∈ N is a sub-coalgebra of (W, ∆ ⊙ , ǫ ⊙ ) of finite graded dimension, so ∆ ⊙ and ǫ ⊙ extend to continuous linear mapsŴ n →Ŵ n⊗Ŵn andŴ n → k, and the following similarly holds: Proposition 3.12. For each n ∈ N, (Ŵ n , ∆ ⊙ , ǫ ⊙ ) is a linearly compact coalgebra.
Nevertheless, there is a sense in which ∇ ¡ and ∆ ⊙ can be interpreted as compatible morphismŝ W⊗Ŵ →Ŵ andŴ →Ŵ⊗Ŵ. This is the main theme of the next section.

Species coalgebroids
Let Mon(C ), Comon(C ), and Bimon(C ) be the categories of monoids, comonoids, and bimonoids in a symmetric monoidal category C . Let FB denote the category of finite sets with bijections as morphisms. A C -species is a functor FB → C . Such functors form a category, denoted C -Sp, with natural transformations as morphisms. For more background on species, see [2,Chapter 8].
When F is a C -species and S is a finite set and σ : S → T is a bijection, we write F [S] for the corresponding object in C and F [σ] for the corresponding morphism F [S] → F [T ], which is necessarily an isomorphism. When η : F → G is a natural transformation and S is a finite set, we write η S for the corresponding morphism With these conventions, a linearly compact coalgebra species is a functor V : FB → Comon( Vec k ). Suppose U , U ′ , V , and V ′ are linearly compact coalgebra species and α : U → U ′ and β : and for each finite set I, where the sums are over all 2 |I| ways of writing I as a union of two disjoint sets.
[J] similarly when σ : I → J is a bijection. The category of linearly compact coalgebra species is symmetric monoidal with respect to this operation, called the Cauchy product in [2], with unit object given by the species 1 : N → Comon( Vec k ) that has 1[∅] = k and 1[S] = 0 for all nonempty finite sets S. When ∇ : V · V → V is a natural transformation and (a) For all pairwise disjoint finite sets S, T , and U , the following diagrams commute: For all disjoint finite sets S and T , the following diagrams commute: For all disjoint finite sets S and T , the following diagram commutes: We refer to ∇ : V · V → V and ι : 1 → V as the product and unit of V , and to the families of maps ∆ = (∆ I ) and ǫ = (ǫ I ) as the coproduct and counit of V . Species coalgebroids form a category, which we denote by Mon(Comon FB ), whose morphisms are the natural transformations between Comon( Vec k )-species that commute with the product and unit morphisms.
for each finite set S and the morphisms ∇ and ι restrict to natural transformations H · H → H and 1 → H . When these conditions hold, we have (H , ∇, ι, ∆, ǫ) ∈ Mon(Comon FB ).
Remark. If needed, one can introduce a sequence of definitions dual to those above. The natural dual of a linearly compact coalgebra species is an algebra species, i.e., a functor FB → Mon(Vec k ). Such functors form a symmetric monoidal category with unit object 1, relative to the Cauchy product defined just as in (4.1) but with the completed tensor product⊗ replaced by ⊗. The natural dual of a species coalgebroid is then a comonoid in the category of algebra species.
Species coalgebroids generalize linearly compact bialgebras since the latter are monoids in the category of linearly compact coalgebras. We highlight three functors to or from Mon(Comon FB ): (i) There is a natural "forgetful" functor for all finite sets S. For each n ∈ N, the symmetric group S n acts as a group of coalgebra and σ ∈ S n is a coideal and we denote the corresponding quotient coalgebra by V [n] Sn = V [n]/I n . Reuse ∆ n and ǫ n to denote the coproduct and counit of V [n] Sn . Consider the compositition where the second arrow is the natural quotient map and the first arrow is the direct sum The [n]-component of ∇ descends to a linear map where Mon(Comon FB fin-dim ) is the full subcategory of finite-dimensional species coalgebroids. The functor K is similar to the bosonic Fock functor defined in [2,Chapter 15].
This means that if [w, λ] ∈ W λ where w = w 1 w 2 · · · w m has m letters, then By Proposition 3.12, W defines a linearly compact coalgebra species FB → Comon( Vec k ). Given disjoint finite sets S and T with n = |S| and m = |T | and bijections (λ, where m is the size of the domain of λ. Finally, let ι ¡ : 1 → W be the natural transformation whose nontrivial component is the linear Remark. We can describe the maps (4.6) and (4.7) more concretely. Let S be a finite set of size n. An S-word is a finite sequence a = a 1 a 2 · · · a l with a i ∈ S. Given a bijection λ : In this way, the product can be defined using the ordinary shuffle operation instead of the shifted shuffle in (2.3).
With slight abuse of notation, we reuse the symbols ∆ ⊙ and ǫ ⊙ to denote the families of maps −→ k for all finite sets S. The following then holds: Proof. Modify the diagrams in

Word relations
Here, we characterize the relations on words that generate sub-objects of the bialgebra W, the linearly compact Hopf algebraŴ P , or the species coalgebroid W . Our starting point is the following: Definition 5.1. A word relation is an equivalence relation ∼ on words with the property that v ∼ w only if v and w share the same set of letters, not necessarily with the same multiplicities.

Algebraic relations
Recall that w ↑ m and w ↓ m are formed from w by adding and subtracting m to each letter.
Definition 5.2. A word relation ∼ is algebraic if for all words v and w, the following holds: Condition (a) states that ∼ is a congruence on the free monoid on P, and is equivalent to requiring that vxw ∼ vyw whenever v, w, x, y are words with x ∼ y. A typical example of an algebraic word relation is K-Knuth equivalence [8,Definition 5.3], the strongest congruence with bac ∼ bca, acb ∼ cab, aba ∼ bab, and a ∼ aa for all integers a < b < c. For this relation, Definition 5.2(b) can be checked directly; see also Proposition 5. 16.
Fix a word relation ∼ and suppose v and w are words. We note two basic facts: Proof. Take m = 0 in condition (b) in Definition 5.2.
Given a set E of words with letters in [n] and a bijection λ : For each finite set S of size n ∈ N, let K defines a subspecies of W .
Proof. The definition of a word relation implies that the empty word ∅ belongs to its own ∼equivalence class, so the element [∅, . This observation shows that ι ¡ always restricts to a natural transformation 1 → K (∼) . By the comments after Defini- and (v ∩ I) ↓ n ∼ (w ∩ I) ↓ n for I = n + P. By taking E and F to be the ∼-equivalence classes of v ∩ [n] and (v ∩ I) ↓ n, one checks that this property is necessary and sufficient to have

Condition (a) in Definition 5.2 holds if and only if ∆
for all disjoint finite sets S and T and basis elements κ λ T . This suffices to show that K (∼) is a sub-coalgebroid if and only if ∼ is algebraic. Continue to let ∼ be a word relation. For n ∈ N, write κ n E in place of κ λ E when λ is the identity map [n] → [n], and let K [n] ∩Ŵ n be the set of elements κ n E where E ranges over all ∼-equivalence classes of words with letters in [n]. Define The vector space K (∼) is a subspace ofŴ but is considered to be a discrete topological space. We say that ∼ is of finite-type if for each n ∈ N, the space K (∼) n is finite-dimensional, or equivalently if the set of words with letters in [n] decomposes as a union of finitely many ∼-equivalence classes.
Proof. If ∼ is algebraic and of finite-type, then the species coalgebroid ( The relation ∼ is homogeneous if v ∼ w implies that v and w have the same length. When this holds, each equivalence class in W n is finite so K (∼) ⊂ W, and each κ n E ∈ K (∼) n is homogeneous.
Proof. The argument is the same as in the proof of Theorem 5.5, mutatis mutandis.
A word of minimal length in its ∼-equivalence class is reduced. A pair [w, n] ∈ W n is reduced with respect to ∼ if w is reduced. Let W

P-algebraic relations
To adapt Theorem 5.5 to packed words, a somewhat technical variation of Definition 5.2 is needed. If u, v ∈ W P are two packed words, then we say that w ∈ W P is a (u, v)-destandardization if there are (not necessarily packed) wordsũ,ṽ such that w =ũṽ and u = fl(ũ) and v = fl(ṽ). For example, 1234, 1324, and 1423 are (12, 12)-destandardizations, as is 1212.
Definition 5.9. A word relation ∼ is P-algebraic if for all v, w ∈ W P , the following holds: Note that property (a) depends on the field k.
The set of packed words W P is a union of equivalence classes under any word relation. Let K (∼) P be the set of sums κ E := w∈E w ∈Ŵ P where E is a ∼-equivalence class in W P . Define The part of the theorem asserting that K (∼) P is a Hopf algebra when ∼ is homogeneous and P-algebraic is formally similar to [19,Theorem 31] and [35, Theorem 2.1], though neither of these results is a special case of our statement, or vice versa.
Proof. We first prove the weaker version of the theorem where both instances of "Hopf subalgebra" are replaced by "sub-bialgebra." Suppose v and w are packed words and E ⊂ W P is a ∼-equivalence P if and only if this coefficient is the same as the corresponding coefficient of v ′ ⊗ w ′ for any packed words v ′ , w ′ with v ∼ v ′ and w ∼ w ′ . It follows thatK (∼) P is a linearly compact sub-coalgebra ofŴ P if and only if condition (a) in Definition 5.9 holds.
One has ∇ ¡ (κ E ⊗ κ F ) ∈K (∼) P for all basis elements κ E , κ F ∈ K (∼) P if and only if condition (b) in Definition 5.9 holds by the same reasoning as in the proof of Theorem 5.5. We conclude that K (∼) P is a linearly compact sub-bialgebra ofŴ P if and only if ∼ is P-algebraic. If ∼ is homogeneous then, in view of Example 3.9, K (∼) P is a graded sub-bialgebra of W P if and only ifK (∼) P is a linearly compact sub-bialgebra ofŴ P .
To upgrade these conclusions to what is stated in the theorem, we first observe that if ∼ is homogeneous and P-algebraic then K (∼) P is a bialgebra that is graded and connected, and all such bialgebras are Hopf algebras [15,Proposition 1.4.16].
Next assume ∼ is P-algebraic but not necessarily homogeneous. ThenK is not just a linearly compact Hopf algebra but a linearly compact Hopf subalgebra of (Ŵ P , ∇ ¡ , ι ¡ , ∆ ⊙ , ǫ ⊙ ), observe that the latter is justK (=) P and is therefore the dual of H (=) , where = is the usual equality relation interpreted as the (P-algebraic) word relation whose equivalence classes all have size one. But H (∼) is evidently a quotient of H (=) , so under dualityK (∼) P becomes a linearly compact Hopf subalgebra ofK Corollary 5.11. If ∼ is P-algebraic and of finite-type then (K This bialgebra is not necessarily graded so may fail to be a Hopf algebra; see Example 6.5. Proof. Assume ∼ is P-algebraic and of finite-type. All products and coproducts of basis elements in K (∼) P are finite sums of (tensors of) other basis elements, so belong to K (∼) An algebraic word relation is not necessarily P-algebraic, or vice versa. The following is a natural sufficient condition for this to occur.
Lemma 5.12. Let ∼ be an algebraic word relation. Assume that whenever v and w are words with the same set of letters and fl(v) ∼ fl(w), it holds that v ∼ w. Then ∼ is P-algebraic.
Proof. Suppose v, w, v ′ , w ′ ∈ W P and v ∼ v ′ and w ∼ w ′ . For any wordṽ with fl(ṽ) = v, there exists a unique wordṽ ′ that has the same set of letters asṽ and satisfies fl(ṽ ′ ) = v ′ , and for this word we haveṽ ∼ṽ ′ . Given a wordw with fl(w) = w, definew ′ analogously. The mapṽw →ṽ ′w′ is then a bijection between the sets of (v, w)-and (v ′ , w ′ )-destandardizations in any ∼-equivalence class, so ∼ is P-algebraic.

Uniformly algebraic relations
Problematically, we do not know of any efficient method to check whether an arbitrary word relation satisfies condition (a) in Definition 5.9, or to generate relations that have this property. It is therefore useful in practice to consider the following less general type of relation: Definition 5.13. A word relation ∼ is uniformly algebraic if for all words v, w, the following holds: If v ∼ w and I ⊂ P is an interval (i.e., a set of consecutive integers), then v ∩ I ∼ w ∩ I.  [14,19] as compatibility with (de)standardization.  for all words a and b, pairs {v, w} ∈ G, and integers 0 ≤ m < min(v) = min(w) is then an algebraic word relation. If it holds that {φ(v), φ(w)} ∈ G whenever {v, w} ∈ G and φ : P → P is an order-preserving injective map, then ∼ is uniformly algebraic.
We refer to ∼ as the strongest algebraic word relation with v ∼ w for {v, w} ∈ G.
Proof. Condition (a) in Definition 5.2 holds if and only if one has avb ∼ awb whenever a, b, v, w are words with v ∼ w, which is evidently the case here. To check condition (b) in Definition 5.2, let I = {m+1, m+2, . . . , n} be an interval in P, fix a pair {v, w} ∈ G, and let 0 ≤ k < min(v) = min(w). It suffices to show thatṽ := ((v ↓ k) ∩ I) ↓ m ∼ ((w ↓ k) ∩ I) ↓ m =:w. Sinceṽ = (v ∩ J) ↓ (m + k) andw = (w ∩ J) ↓ (m + k) for J = k + I, and since we know that either v ∩ J = w ∩ J or {(v ∩ J) ↓ l, (w ∩ J) ↓ l} ∈ G for an integer 0 ≤ l ≤ m + k, the desired conclusion follows. Now assume that {φ(v), φ(w)} ∈ G whenever {v, w} ∈ G and φ : P → P is an order-preserving injective map. To show that ∼ is uniformly algebraic, it suffices by Lemma 5.14 to check that φ(x) ∼ φ(y) whenever x and y are words with x ∼ y and φ : P → P is an order-preserving injection. It is enough to show this when x = a(v ↓ m)b and y = a(w ↓ m)b for some {v, w} ∈ G, where 0 ≤ m < min(v) = min(w) and where a and b are arbitrary words. Observe that φ(v ↓ m) = ψ(v) ↓ m and φ(w ↓ m) = ψ(w) ↓ m where ψ : P → P is the map with This map is an order-preserving injection, so we have {ψ(v), ψ(w)} ∈ G by hypothesis, and it also holds that 0 ≤ m < min(ψ(v)) = min(ψ(w)). Thus holds by the definition of ∼, as desired.
The following example is instructive when comparing the definitions in this section. Let (W, S) be a Coxeter system with length function ℓ : W → N. There exists a unique associative product for the words v ′ = ababa · · · (n − 1 letters) and w ′ = babab · · · (n − 1 letters).

This means that if
• ∼ is uniformly algebraic then m(i, j) = m(i + 1, j + 1) for all i, j ∈ P.
Proof. Combining Lemmas 5.4, 5.14, and 5.17 shows that the given conditions are necessary. Condition (a) in Definition 5.2 holds for • ∼ by construction. Assume m(i, j) ≤ m(i + 1, j + 1) for all i, j ∈ P and let I = [k + 1, n] for some k, n ∈ N. To check condition (b) in Definition 5.2, it suffices to show that if v = ababa · · · and w = babab · · · for some a, b ∈ P, where both words have m(a, b) letters, then (v ∩ I) ↓ k • ∼ (w ∩ I) ↓ k. This is clear when I ∩ {a, b} = {a, b} and holds when {a, b} ⊂ I by Lemma 5.17. Thus • ∼ is algebraic. It follows by Lemmas 5.14 and 5.17 that the condition for • ∼ to be uniformly algebraic is also sufficient.
A generator s i belongs to the center Z(W ) of W if and only if m(i, j) = 2 for all j ∈ P \ {i}. The group W is abelian if and only if W = Z(W ), which occurs when m(i, j) = 2 for all i < j.
Proposition 5.19. If W is abelian, then • ∼ is uniformly algebraic and of finite-type. If W is non-abelian and p ∈ P is minimal such that s p / ∈ Z(W ), then • ∼ is algebraic and of finite-type if and only if for some q ∈ P it holds that m(i, i + q) = 3 and m(i, j) = 2 for all p ≤ i < j = i + q. If these conditions hold, then the word relation • ∼ is uniformly algebraic when p = q = 1 but not P-algebraic over any field when p > 1 or q > 1.
We discuss the bialgebras K ( • ∼) and K Proof. The proof depends on the classification of finite Coxeter groups. The • ∼-equivalence classes in W n are in bijection with the elements of the parabolic subgroup s 1 , s 2 , . . . , s n ⊂ W , so • ∼ is of finite-type if and only if each of these subgroups is finite. In the listed cases, each subgroup of this form is a finite direct product of finite symmetric groups, and is therefore finite.
If W is abelian then both conditions in Proposition 5.18 obviously hold, so • ∼ is uniformly algebraic. Assume W is non-abelian and we have m(i, i + q) = 3 and m(i, j) = 2 for all p ≤ i < j = i + q, where p ∈ P is minimal with s p / ∈ Z(W ). The first condition in Proposition 5.18 is clear, and the second condition holds if and only if p = q = 1. Hence • ∼ is uniformly algebraic when W is non-abelian if and only if p = q = 1. Assume instead that p > 1 or q > 1. In this case we have 12 • ∼ 21, but the • ∼-equivalence class of the 2-letter word p(p + q) consists of all words of the form pp · · · p(p + q)(p + q) · · · (p + q) and so contains exactly one (12, ∅)-destandardization and no (21, ∅)-destandardizations. Thus condition (a) in Definition 5.9 fails so • ∼ is not P-algebraic. Continue to assume W is non-abelian and p ∈ P is minimal with s p / ∈ Z(W ). Suppose • ∼ is algebraic and of finite-type, so that m(i, j) ≤ m(i + 1, j + 1) for all i, j ∈ P. We cannot have m(i, j) > 3 for any i < j since then 3 < m(j, 2j − i) and s i , s j , s 2j−i would be infinite. Since s p / ∈ Z(W ) but {s 1 , s 2 , . . . , s p−1 } ⊂ Z(W ), there exists a minimal q ∈ P such that m(p, p + q) = 3. Then m(i, i + q) = 3 for all i ≥ p. We cannot have m(i, j) = 3 for any p ≤ i < j = i + q as then we would also have m(i + q, j + q) = 3 so the Coxeter graph of (W, S) would contain a cycle and some s 1 , s 2 , . . . , s n would be infinite. Hence m(i, i + q) = 3 and m(i, j) = 2 for all p ≤ i < j = i + q.

Examples
This section presents some further examples of word relations and related bialgebras. Let H n ∈ K (∼) P denote the n-letter packed word 111 · · · 1, so that H 0 = ∅ is the unit element in K (∼) P . Each H n is homogeneous of degree n, and the algebra structure on K (∼) P is just the polynomial algebra k H 1 , H 2 , . . . where H 1 , H 2 , . . . are interpreted as non-commuting indeterminates. The coproduct of K (∼) P satisfies ∆ ⊙ (H n ) = n i=0 H i ⊗ H n−i . This graded Hopf algebra is commonly known as the algebra of noncommutative symmetric functions NSym [12] or Leibniz-Hopf algebra.
Example 6.2. Define K-equivalence to be the strongest algebraic word relation with a ∼ aa for all a ∈ P. This is the case of the relation • ∼ described in the previous section when (W, S) is a universal Coxeter, i.e., when m(i, j) = ∞ for all i < j. K-equivalence is therefore uniformly algebraic but neither homogeneous nor of finite-type. One has v ∼ w if and only if v and w coincide after all adjacent repeated letters are combined.
Each equivalence class under ∼ contains a unique reduced word with no equal adjacent letters, which we call a partial (small) multi-permutation. A (small) multi-permutation is a partial multipermutation that is also a packed word. This notion of a multi-permutation is what is intended in [24, Definition 4.1], which omits our condition about being a packed word (and so inadvertently gives the definition of a partial multi-permutation).
For a partial multi-permutation w with max(w) n . Given an arbitrary list w 1 , w 2 , . . . of distinct partial multi-permutations with letters in [n] and coefficients c 1 , c 2 , · · · ∈ k, we abbreviate our notation by setting If v and w are partial multi-permutations with letters in [m] and [n], respectively, then where ⋆ is the multishuffle product described by [24,Proposition 3.1], while where is the cuut coproduct defined in [24, §3]. From (4.6) and (4.7), these formulas completely determine the (co)product of the species coalgebroid ( The linearly compact Hopf algebra (K Example 6.3. Define the K-commutation relation to be the transitive closure ∼ of K-equivalence and the commutation relation. This is the weakest word relation, in the sense that any word relation is a subrelation of ∼. As the relation ∼ is the special case of • ∼ when W is abelian, it is uniformly algebraic, inhomogeneous, and of finite-type by Proposition 5. 19. The ∼-equivalence classes in W n are in bijection with subsets I ⊂ [n]. All packed words w with max(w) = n belong to the same ∼-equivalence class. If we let κ n ∈ K (∼) P denote the sum of these words, then κ n = ∇ (n−1) ¡ (x ⊗ x ⊗ · · · ⊗ x) for x = κ 1 = 1 + 11 + 111 + . . . . Thus K (∼) P coincides as an algebra with k[x], but its coproduct has ∆ ⊙ (x) = x ⊗ 1 + x ⊗ x + 1 ⊗ x. This is the q = 1 version of the univariate infiltration bialgebra discussed, for example, in [ Suppose λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ m > 0) is an integer partition and w = w 1 w 2 · · · w m is the factorization of a word w into maximal weakly increasing subwords. Slightly abusing standard terminology, say that w is a semistandard tableau of shape λ if ℓ(w i ) = λ m+1−i for i ∈ are noted in [25,37]. The subalgebra of K for all integers a < b < c. Proposition 5.16 implies that this relation is uniformly algebraic. Though less well-studied than its homogeneous analogue, K-Knuth equivalence appears to be an equally fundamental case of interest. Its relationship with Hecke insertion [7] is parallel to that of Knuth equivalence with the RSK correspondence. Again with minor abuse of standard terminology, define an increasing tableau to be a semistandard tableau with no equal adjacent letters, i.e., in which every weakly increasing consecutive subword is strictly increasing. For example, the words ∅ or 645123 or 5612 or 545234 are all increasing tableaux under our definition.
There are finitely many increasing tableaux with all letters in a given finite set [37, Lemma 3.2] and every K-Knuth equivalence class contains at least one increasing tableau [13,Lemma 58]. Thus, K-Knuth equivalence is of finite-type and a somewhat improved way of indexing the elements of K (∼) n is to define [[T, n]] = w∼T [w, n] for each increasing tableau T with max(T ) ≤ n. The usefulness of this construction is limited, since it is not known how to easily detect when two increasing tableaux are K-Knuth equivalent. There is an algorithm to compute all K-Knuth classes of words with a given set of letters, however [13].
It is an open problem to find an irredundant indexing set for K-Knuth equivalence classes, with respect to which one can describe explicitly the product and coproduct of the bialgebras K (∼) and K (∼) P . Patrias and Pylyavskyy [37] refer to the latter as the K-theoretic Poirier-Reutenauer bialgebra KPR. They note that KPR is not a Hopf algebra [37, §4] and give some (necessarily inexplicit) formulas for its product and coproduct; see [37,Theorems 4.3,4.5,4.10,and 4.12]. Theorem 5.10 for K-Knuth equivalence recovers [37,Theorem 4.15].
As noted in [8,Remark 5.10] and [13, §4], the set of reduced words in a K-Knuth equivalence class may fail to be spanned by the homogeneous relations bac ∼ bca, acb ∼ cab, and aba ∼ bab for a < b < c. The graded sub-bialgebra K (∼) R ⊂ K (∼) is thus in some sense not any easier to study. We mention one other property of this relation. Define weak K-Knuth equivalence to be the Let w r be the word obtained by reversing w. If v ≈ w then v r v ∼ w r w since baab ∼ bab ∼ aba ∼ abba. Buch and Samuel state the converse as [8,Conjecture 7.10], which appears to be still unresolved: Conjecture 6.6 (Buch and Samuel [8]). Two words v and w are weakly K-Knuth equivalent if and only if v r v and w r w are K-Knuth equivalent.
Example 6.7. One avoids many pathologies of K-Knuth equivalence by considering the stronger relation of Hecke equivalence, which is the strongest algebraic word relation ∼ with ac ∼ ca, aba ∼ bab, and a ∼ aa for all positive integers a < b < c, so that 13 ∼ 31 but 12 ∼ 21 [8,Definition 6.4]. As explained in Proposition 5.19, this is the only case of the relation • ∼ that is uniformly algebraic and of finitetype for which the ambient Coxeter group is non-abelian. Each set K (∼) n is in bijection with the symmetric group S n+1 , which we view as the set of words of length n + 1 containing each i ∈ [n + 1] as a letter exactly once.
Given π ∈ S n+1 , let [[π]] = w [w, n] ∈ K (∼) n denote the sum over Hecke words for π, i.e., words w = w 1 w 2 · · · w m with π = s w 1 • s w 2 • · · · • s wm where • is the product defined in Section 5.3 and s a = (a, a + 1) ∈ S n+1 . The coproduct of We have a better understanding of the graded bialgebra of reduced classes K (∼) R when ∼ is Hecke equivalence. This bialgebra is the main topic of our complementary paper [32], which derives a recursive formula for the product of any two basis elements in K The analogue of Conjecture 6.6 for Hecke equivalence is known to be true: Example 6.9. Fix integers p, q ∈ P and define ≈ to be the strongest algebraic word relation with (i) a(a + q)a ≈ (a + q)a(a + q) for all integers a ≥ p, (ii) ab ≈ ba for all positive integers a < b with a < p or b = a + q, and (iii) a ≈ aa for all positive integers a.
When p = q = 1 this relation coincides with Hecke equivalence from Example 6.7. If min{p, q} > 1 then ≈ corresponds to the cases of • ∼ in Proposition 5.19 that are algebraic and of finite-type but not P-algebraic. In the latter situation K (≈) is a bialgebra, but K (≈) P is not a sub-bialgebra of W P . If p = 1 and we write ∼ for Hecke equivalence, then the subspace (q) K (≈) := r∈N K (≈) Example 6.10. Let ∼ be the transitive, reflexive closure of the relation that satisfies puvuq ∼ puvq for all words p, q, u, and v.
We refer to this relation as left-regular band (LRB) equivalence. It is straightforward to check that LRB equivalence is a uniformly algebraic word relation of finite-type. The reduced words for this relation are the words with all distinct letters, often called injective words or partial permutations. Distinct reduced words for LRB equivalence are never equivalent. The quotient of the free monoid by ∼ is the free left-regular band discussed, for example, in [5, §1.3]. The packed injective words are precisely the permutations of [n] for all n ∈ N, which index a basis for K (∼) P . By considering this basis, it is easy to see that the algebra structures on K (∼) P and the Malvenuto-Poirier-Reutenauer Hopf algebra FQSym mentioned at the end of Section 2.3 are isomorphic. The coproduct for K (∼) P is more complicated than for FQSym, however, and is no longer graded.
We mention one other miscellaneous example which will be of significance in Section 8.

Combinatorial bialgebras
A composition α of n ∈ N, written α n, is a sequence of positive integers α = (α 1 , α 2 , . . . , α l ) with α 1 + α 2 + · · · + α l = n. The nonzero numbers α i are the parts of the composition. The unique composition of n = 0 is the empty word ∅. Let k[[x 1 , x 2 , . . . ]] be the algebra of formal power series with coefficients in k in a countable set of commuting variables. The monomial quasi-symmetric function M α indexed by a composition α n with l parts is When α is the empty composition, set M ∅ = 1. Each α n can be rearranged to form a partition of n, denoted sort(α). The monomial symmetric function indexed by a partition λ is m λ = sort(α)=λ M α . Write λ ⊢ n when λ is a partition of n and let Sym n = k-span{m λ : λ ⊢ n}. The subspace Sym = n∈N Sym n ⊂ QSym is the familiar graded Hopf subalgebra of symmetric functions.
Let NSym = k H 1 , H 2 , . . . be the graded Hopf algebra of noncommutative symmetric functions described in Example 6.1, that is, the k-algebra of polynomials in non-commuting indeterminates H 1 , H 2 , H 3 , . . . , where H n has degree n and the coproduct has ∆(H n ) = n i=0 H i ⊗ H n−i . Given α n with l parts, let H α = H α 1 H α 2 · · · H α l and define H ∅ = H 0 = 1. NSym is the graded dual of QSym via the bilinear form NSym × QSym → k in which {H α } and {M α } are dual bases [1, §3].
If ζ : V → k[t] is a map and a ∈ k, then let ζ| t=a : V → k be the map v → ζ(v)(a).
A combinatorial Hopf algebra is a combinatorial bialgebra (V, ∇, ι, ∆, ǫ, ζ) in which (V, ∇, ι, ∆, ǫ) is a Hopf algebra. These definitions are minor generalizations of the notions of combinatorial coalgebras and Hopf algebras in [1], where it is required that V have finite graded dimension and dim V 0 = 1.
When the structure maps are clear from context, we refer to just the pair (V, ζ) as a combinatorial coalgebra or bialgebra. A morphism φ : (V, ζ) → (V ′ , ζ ′ ) of combinatorial coalgebras or bialgebras is a graded coalgebra or bialgebra morphism φ : V → V ′ satisfying ζ ′ = ζ • φ. The map ζ is the character of a combinatorial coalgebra or bialgebra (V, ζ).
Remark. Specifying a graded linear map (respectively, algebra morphism) V → k[t] is equivalent to defining a (multiplicative) linear map V → k. We define the character ζ to be a map V → k[t] since this extends more naturally to the linearly compact case. This convention differs from [1,32], where the character of a combinatorial coalgebra is defined to be a linear map V → k. Example 7.3. There is a graded algebra morphism ζ QSym : QSym → k[t] that has ζ QSym (M ∅ ) = 1, ζ QSym (M (n) ) = t n for each n ≥ 1, and ζ QSym (M α ) = 0 for all other compositions α. One way to see that the graded linear map ζ QSym is an algebra morphism is to observe that it is the restriction of the algebra morphism k[[x 1 , x 2 , . . . ]] → k[[t]] that sets x 1 = t and x n = 0 for all n > 1. The pair (QSym, ζ QSym ) is a combinatorial Hopf algebra.
Proof. This result is only slightly more general than [1, Theorem 4.1] and has essentially the same proof. We sketch the argument. Let NSym denote the completion of NSym with respect to the basis {H α }. Since NSym is a graded algebra, NSym is a linearly compact algebra. Write ·, · for both the tautological form V × V * → k and the bilinear form QSym × NSym → k, continuous in the second coordinate, relative to which the pseudobasis {H α } ⊂ NSym is dual to the basis {M α } ⊂ QSym. Both forms are nondegenerate. We view the dual space V * as the linearly compact algebra with unit element ǫ dual to the coalgebra V via the tautological form. The linearly compact algebra structure on NSym is the one dual to the coalgebra structure on QSym. Let [t n ]f denote the coefficient of t n in f ∈ k[t] and define ζ n ∈ V * by ζ n (v) = [t n ]ζ(v), so that ζ 0 = ǫ. Observe that [t n ]ζ QSym (q) = q, H n for all n ∈ N and q ∈ QSym. It follows that there exists a unique coalgebra morphism ψ : V → QSym with ζ = ζ QSym • ψ if and only if there exists a unique linearly compact algebra morphism φ : NSym → V * with φ(H n ) = ζ n for all n ∈ N, and when this occurs, the two maps satisfy ψ(v), w = v, φ(w) for all v ∈ V and w ∈ NSym.
Write V * gr for the graded algebra that is the graded dual of the graded coalgebra V . Since the unit of V * gr is ǫ = ζ 0 , there is a unique algebra morphism NSym → V * gr that sends H n → ζ n for each n ∈ N. As V m ⊂ ker ζ n for all m = n, this morphism is graded, so it extends to a unique linearly compact algebra morphism φ : NSym → V * . The resulting morphism φ : NSym → V * is evidently the unique one satisfying φ(H n ) = ζ n for all n ∈ N, and one has φ(H α ) = ζ α with ζ α as in (7.1). Hence, there exists a unique coalgebra morphism ψ : V → QSym satisfying ζ = ζ QSym • ψ, and for this map one has ψ(v), H α = v, φ(H α ) = v, ζ α = ζ α (v) for all v ∈ V and compositions α; in other words, ψ is the graded linear map (7.1). 1 Assume (V, ζ) is a combinatorial bialgebra. Use the symbol ∇ to also denote the products of k[t] and QSym. Define ξ = ∇ • (ζ ⊗ ζ). Then (V ⊗ V, ξ) is a combinatorial coalgebra and it is easy to check that ∇ • (ψ ⊗ ψ) and ψ • ∇ are both morphisms (V ⊗ V, ξ) → (QSym, ζ QSym ). The uniqueness proved in the previous paragraph implies that ∇ • (ψ ⊗ ψ) = ψ • ∇. Since ζ is an algebra morphism, we also have ψ(1) = 1 ∈ QSym, so ψ is a bialgebra morphism. We often refer to just the pair (V, ζ) as a linearly compact combinatorial coalgebra or bialgebra. The map ζ is the character of (V, ζ). A morphism φ : (V, ζ) → (V ′ , ζ ′ ) of linearly compact combinatorial (co/bi)algebras is a continuous (co/bi)algebra morphism satisfying ζ ′ = ζ • φ. for m > 2. Let ζ ∅ := ζ| t=0 = ǫ. Given α n > 0, let ζ α : V → k be the map whose value at v ∈ V is the coefficient of t α 1 ⊗ t α 2 ⊗ · · · ⊗ t αm in the image of v under Define ψ : V → QSym to be the map where the sum is over all compositions α. This is the same formula as (7.1), except now the sum may have infinitely many nonzero terms.
Proof. The proof is similar to that of Theorem 7.4. Write ·, · for both the tautological form V ∨ × V → k and the bilinear form NSym × QSym → k, continuous in the second coordinate, relative to which the pseudobasis {M α } ⊂ QSym is dual to the basis {H α } ⊂ NSym. Both forms are nondegenerate. We view the vector space V ∨ of continuous linear maps V → k as the algebra with unit element ǫ dual to the linearly compact coalgebra V via the tautological form. The algebra structure on NSym is the one dual to the linearly compact coalgebra structure on QSym.
The preceding result removes the requirement of a grading in Theorem 7.4, at the cost of working with linearly compact spaces. If one needs to work with honest (bi, co)algebras, then it is still possible to remove the requirement of a grading in Theorem 7.4, but then one must impose a technical finiteness condition to ensure that the sum (7.1) belongs to QSym.
This definition is similar to the notion of a combinatorial Hopf monoid given in [31, §5.4]. As usual, when the other data is clear from context, we refer to just (V , Z) as a combinatorial coalgebroid. The natural transformation Z : If (V, ζ) is a linearly compact combinatorial bialgebra, then (E(V ), E(ζ)) is a combinatorial coalgebroid. Let E QSym = E( QSym) and Z QSym = E(ζ QSym ). Suppose (V , Z) is a combinatorial coalgebroid. Define Ψ : V → E QSym to be the natural transformation such that, for each set S, is the product of k[t]. We refer to X(V ) as the character monoid of V . If V is a Hopf algebra with antipode S, then ζ −1 := ζ • S is the left and right inverse of ζ ∈ X(V ), and X(V ) is a group with some notable properties [1].
If (V, ∇, ι, ∆, ǫ) is a linearly compact k-bialgebra then we let X(V ) denote the set of continuous linear maps ζ : V → k[[t]] for which (V, ζ) is a linearly compact combinatorial bialgebra. This set is again a monoid with unit element ǫ and product ζζ ′ :

Characters and morphisms
In this section, we assume k has characteristic zero and view W as the bialgebra from Theorem 2.4. Our goal here is to illustrate a variety of cases where well-known symmetric and quasi-symmetric functions may be constructed via the morphisms in Theorems 7.4 and 7.8 and Corollary 7.10.

Fundamental quasi-symmetric functions
We start by examining four natural elements of X(W). Let ζ ≤ : W → k[t] be the linear map Define ζ ≥ , ζ < , ζ > to be the linear maps W → k[t] given by the same formula but with "weakly increasing" replaced by "weakly decreasing," "strictly increasing," and "strictly decreasing." Proof. This is equivalent to [32,Proposition 5.4] and easily checked directly.
If ∼ is an algebraic word relation so that K (∼) R ⊂ W is a graded sub-bialgebra, then ζ • restricts to an element of X(K
Given finite, nonempty subsets S, T ⊂ P, write S T if max(S) ≤ min(T ) and S ≺ T if max(S) < min(T ), and define x S = i∈S x i . In [24, §5.3], Lam and Pylyavskyy define the multifundamental quasi-symmetric function of a composition α n to be the power series where the sum is over finite, nonempty sets S 1 , S 2 , . . . , S n of positive integers.
, then we use the shorthand f ( x 1−x ) to denote the power series obtained from f by substituting Recall that a multi-permutation is a packed word with no adjacent repeated letters. The functionsL α arise naturally as the images of the pseudobasis of the Hopf algebra mMR =K (∼) P when ∼ is K-equivalence, under the morphisms (mMR,ζ • ) → ( QSym, ζ QSym ).
The first identity is equivalent to [24,Theorem 5.11].
Proof. Assume w = w 1 w 2 · · · w n has n letters and let m 1 , m 2 , . . . , m n ∈ P. The first identity holds since, by Proposition 8.2,ψ < applied to the word (w 1 w 1 · · · w 1 )(w 2 w 2 · · · w 2 ) · · · (w n w n · · · w n ) ∼ w with each w i repeated m i times gives the sum in (8.2)  If ∼ is an algebraic word relation so that K (∼) ⊂ W is sub-coalgebroid, then the natural transformation Z • restricts to an element of X(K (∼) ) and Ψ • restricts to the unique morphism of combinatorial coalgebroids (K (∼) , Z • ) → (E QSym, Z QSym ).
Example 8.11. Again suppose ∼ is the commutation relation from Example 6.1, so that we can identify NSym ∼ = K (∼) P by setting H n = 11 · · · 1 ∈ K (∼) P . The characterζ >|≤ corresponds to the algebra morphism NSym → k[t] with H n → 2t n for n > 0, and we havẽ ψ >|≤ (H n ) = K (n) = α n 2 ℓ(α) M α = λ⊢n 2 ℓ(λ) m λ = q n where q n ∈ Sym is the symmetric function such that n≥0 q n t n = i≥1  gives the Stanley symmetric functions F π and F C π of types A and C; see the discussion in [32].

Symmetric functions
Suppose ∼ is a uniformly algebraic word relation. It is natural to ask when the image ofK Moreover, if these conditions hold and E is any ∼-equivalence class of packed words, then the symmetric functionsψ ≤ (κ E ) andψ > (κ E ) are both Schur positive.
There is a left-handed version of this result, in which the symbols ≤ and > are replaced by ≥ and <, and Knuth equivalence in part (c) is replaced by reverse Knuth equivalence: the relation with v ∼ w if and only if v r and w r are Knuth equivalent. One can ask similar questions about (P-)algebraic word relations, but such relations do not seem to have a nice classification.
Proof. The continuous linear map QSym → QSym with L α → L α c restricts to the continuous linear involution of Sym with s λ → s λ T , so parts (a) and (b) are equivalent by Proposition 8.2.
By Proposition 8.10, the imageψ >|≤ K (∼) P is contained in the completion of OQSym with respect to its basis of peak quasi-symmetric functions {K α }. By [46,Theorem 3.8], the intersection of this completion with Sym is the linearly compact space of formal power series k[[q 1 , q 2 , q 3 , . . . ]], which is also the completion of OSym with respect to its basis of Schur Q-functions.
It follows from Examples 8.12 and 8.9 that if ∼ extends Knuth equivalence or K-Knuth equivalence thenψ >|≤ (κ) is Schur Q-positive for all elements κ ∈ K (∼) P . If we could prove the following, then we could upgrade the "only if" in Proposition 8.16 to "if and only if." Conjecture 8.17. If ∼ is exotic Knuth equivalence, thenψ >|≤ (κ) ∈ Sym for κ ∈ K (∼) P .
An even stronger property appears to be true: Curiously,ψ >|≤ (κ) is not always Schur Q-positive when κ ∈ K (∼) P and ∼ is exotic Knuth equivalence. We have checked the two conjectures when κ = κ E where E is any exotic Knuth equivalence class of words of length at most nine. Among the 27,021 classes E of packed words w with ℓ(w) = 9, only 35 are such thatψ >|≤ (κ E ) is not Schur Q-positive.